IN   MEMORIAM 
FLORIAN  CAJORl 


HIGH   SCHOOL  ALGEBRA 


BY 


J.    H.    TANNER,   Ph.D. 

PROFESSOR  OF  MATHEMATICS  IN   CORNELL  UNIVERSITY 


NEW  YORK  .:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


The  Modern  Mathematical  Series, 
lucien  augustus  wait, 

(^Senior  Professor  of  Mathematics  in  Cornell  University,) 
GENERAL   EDITOR. 


This  series  includes  the  following  works : 
ANALYTIC  GEOMETRY.    By  J.  H.  Tanner  and  Joseph  Allen. 
DIFFERENTIAL   CALCULUS.     By  James  McMahon  and  Virgil  Snyder. 
INTEGRAL   CALCULUS.    By  D.  A.  Murray. 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS.     By  Virgil  Snyder  and  J. 
Hutchinson. 


HIGH  SCHOOL  ALGEBRA.  By  J.  H.  Tanner. 
ELEMENTARY  ALGEBRA.  By  J.  H.  Tanner. 
ELEMENTARY  GEOMETRY.    By  James  McMahon. 


The  High  School  Algebra  and  the  Elementary  Algebra  cover  substantially  the 
same  ground :  each  of  them  is  designed  to  meet  college  entrance  requirements  in 
elementary  algebra ;  the  High  School  Algebra,  however,  presents  the  briefer  and 
simpler  treatment  of  the  two. 


COi'VKlGHT,    190T,    BY 

J.   H.   TANNER. 


TANNBB'8   HIGH   8CH.   ALG. 
W.  P.  I. 


-3+ 


PREFACE 

In  the  preparation  of  this  book  the  author's  aim  has  been : 

(1)  To  make  the  transition  from  arithmetic  to  algebra  as  easy 
and  natural  as  possible,  and  to  arouse  the  pupil's  interest  by 
showing  him  early  some  of  the  advantages  of  algebra  over 
arithmetic. 

(2)  To  present  the  several  topics  in  the  order  of  their  sim- 
plicity, giving  definitions  only  where  they  are  needed,  and  insur- 
ing clearness  of  comprehension  by  an  abundance  of  concrete 
illustrations  and  inductive  questions. 

(3)  To  provide  a  large,  well-chosen,  and  carefully  graded  set  of 
exercises,  the  solution  of  which  will  help  not  only  to  fix  in  the 
pupil's  mind  the  principles  involved,  but  also  further  to  unfold 
those  principles. 

(4)  To  omit  non-essentials,  and  yet  provide  a  book  that  fully 
meets  the  entrance  requirements  in  elementary  algebra  of  any 
college  or  university  in  this  country. 

Among  other  features  of  this  book  to  which  attention  is  in- 
vited are:  (1)  the  careful  statement  of  definitions  and  principles; 
(2)  the  emphasis  laid  upon  translating  formulas  and  equations 
into  verbal  language,  and  vice  versa;  (3)  the  inclusion  of  many 
formulas  from  physics  which  the  pupils  are  asked  to  solve  for 
the  various  letters  which  they  contain;  and  (4)  the  extensive 
cross-references,  as  well  as  the  many  "  hints  "  and  "  suggestions  " 
found  among  the  exercises  and  problems,  all  calculated  to  throw 
sidelights  upon  the  work. 

On  the  request  of  several  prominent  mathematics  teachers  the 
author  has  put  an  elementary  chapter  on  quadratic  equations 
(Chap.  XII)  before  the  chapters  on  radicals,  imaginaries,  and  the 
theory  of  exponents.  This  arrangement  is  made  possible  by  the 
treatment  of  radicals  of  the  second  order  given  in  §  121  (only  such 


iv  PREFACE 

radicals  are  met  with,  in  Chap.  XII),  and  it  has  important  peda- 
gogical as  well  as  practical  advantages  over  the  more  usual 
arrangement.  Those  teachers,  however,  who  prefer  the  usual 
order,  may  omit  §  121  altogether,  and  take  Chaps.  XIV  and  XV, 
except  §§  162,  163,  and  170,  before  taking  Chap.  XII. 

In  order  to  avoid  unnecessary  repetition,  the  work  on  graphs 
has  nearly  all  been  collected  into  a  single  chapter.  This  arrange- 
ment has  made  it  possible  to  give  this  topic  a  somewhat  more 
adequate  treatment  than  is  usual  in  a  book  of  this  kind,  and  to 
do  so  without  giving  it  more  than  its  rightful  amount  of  space. 
By  this  arrangement  also  those  schools  w^hich  do  not  take  graphs 
in  their  first  year's  work  will  find  their  algebra  work  unin- 
terrupted, while  appropriately  .placed  footnotes  indicate  the 
connection  in  which  the  parts  of  this  chapter  may  be  most  advan- 
tageously read  by  those  who  wish  to  include  graphs. 

Scattered  through  the  book  are  a  few  articles  (marked  with 
a  *)  which,  should  be  taken  in  connection  with  the  review  work 
when  time  permits ;  the  omission  of  these  articles,  however,  does 
not  anywhere  break  the  continuity  of  the  work.  For  the  benefit 
of  the  brightest  pupils  in  a  class  there  have  been  inserted  here 
and  there  references  to  the  author's  Elementary  Algebra,  where 
the  topics  concerned  are  discussed  somewhat  more  fully ;  a  few 
copies  of  the  Elementary  Algebra  placed  in  the  school  library  can 
in  this  way  be  made  to  serve  a  very  useful  purpose. 

It  is  with  great  pleasure  that  the  author  acknowledges  his  in- 
debtedness to  the  many  experienced  teachers  of  Algebra  in  High 
Schools  and  Academies  in  all  quarters  of  our  country  whose  sug- 
gestions to  him  have  added  so  much  of  value  to  this  book.  Special 
acknowledgments  are  due  to  Prof.  J.  M.  McPherron  of  the  Los 
Angeles  High  Schools  for  reviewing  the  manuscript  before  it 
went  to  press,  and  to  Miss  Cora  Strong  of  the  State  Normal 
School  of  Greensboro,  N.C.,  who  secured  a  leave  of  absence  from 
her  school  so  that  she  might  give  her  entire  time  to  assisting  in 
the  work  on  this  book ;  to  her  mainly  belongs  the  credit  for  the 
excellent  exercises  which  are  found  in  the  following  pages. 


CONTENTS 

[See  Index  in  back  of  book  for  particular  topics.] 

CHAPTER  PAGE 

I.     Introduction 

I.     Literal  Numbers        . 1 

II.     Elementary  Operations 8 

11.     Positive  and  Negative  Numbers 16 

III.  Addition  and  Subtraction  —  Parentheses' 

I.     Addition 27 

II.     Subtraction 32 

III.     Parentheses 35 

IV.  Multiplication  and  Division 

I.     Multiplication    .        .         .        .        .        .        .        .38 

11.     Division 45 

V.     Equations  and  Problems 

Review  Exercise  —  Chapters  I-V 67 

VI.     Type  Forms  in  Multiplication  —  Factoring 

I.     Some  Type  Forms  in  Multiplication         ...      71 
11.     Factoring 77 

VII.     Highest  Common  Factor  —  Lowest  Common  Multiple 

I.     Highest  Common  Factor 98 

II.     Lowest  Common  Multiple 105 

VIII.     Algebraic  Fractions 109 

IX.     Simple  Equations 125 

X.     Simultaneous  Simple  Equations 

I.     Two  Unknown  Numbers 144 

11.     Three  or  More  Unknown  Numbers  ....     156 
Review  Exercise  —  Chapters  VI-X    .         .        .         .         .166 

V 


VI 


CONTENTS 


C^KVT^^  PAGE 

XI.    Involution  and  Evolution 

I.     Involution 170 

II.     Evolution 176 

XII.     Quadratic  Equations  (Elementary) 

I.     Equations  in  One  Unknown  Number    .         .     '   .  190 

II.     Simultaneous  Equations  involving  Quadratics      .  206 

XIII.  Graphic  Representation  of  Equations      .        .        .  219 

XIV.  Irrational  Numbers  —  Radicals 234 

XV.    Imaginary  Numbers 257 

XVI.     Theory  of  Exponents  (Zero,  Negative,  and  Fractional 

Exponents) 265 

XVII.     Quadratic  Equations  (Supplementary  to  Chapter  XII)  277 

Review  Exercise  —  Chapters  XI-XVII   ....  284 

XVIII.     Inequalities 287 

XIX.    Ratio,  Proportion,  and  Variation 

I.     Ratio 293 

II.     Proportion 295 

III.     Variation          . 301 

XX.     Series  —  The  Progressions 

I.     Arithmetical  Progression 307 

II.     Geometric  Progression 313 

XXI.     Mathematical  Induction  —  Binomial  Theorem        .  321 

XXII.    Logarithms 329 

Table  of  Common  Logarithms 341 

INDEX 343 


HIGH   SCHOOL  ALGEBRA 


•       CHAPTER   I 
INTRODUCTION 
I.   LITERAL   NUMBERS 

1.  Algebra.  In  the  following  pages,  we  shall  continue  to 
use  the  symbols  0,  1,  2,  3,  etc.  to  represent  numbers,  and 
the  signs  +,  — ,  x,  -^,  and  =,  to  denote  addition,  subtrac- 
tion, multiplication,  etc.;  that  is,  we  shall  use  all  of  these 
characters  just  as  we  used  them  in  arithmetic.  We  shall 
presently  see,  however,  that  algebra  greatly  simplifies  the 
solution  of  certain  kinds  of  problems  (§  3),  that  it  introduces 
new  kinds  of  number  (§§  13,  146,  164),  and  that  it  makes 
extensive  use  of  letters  to  represent  numbers. 

2.  Numbers  represented  by  letters.  In  arithmetic,  num- 
bers are  almost  always  expressed  by  means  of  the  symbols 
0,  1,  2,  3,  etc.,  but  letters  also  are  sometimes  used.  For 
example,  in  interest  problems  p  often  stands  for  principal^ 
r  for  rate^  t  for  time^  i  for  interest^  and  a  for  amount. 

In  algebra,  on  the  other  hand,  the  use  of  letters  to  repre- 
sent numbers  is  very  common ;  thus,  just  as  in  arithmetic 
we  speak  of  4  books,  7  bicycles,  85  pounds,  3  men,  etc.,  so 
in  algebra  we  use,  not  only  these  expressions,  but  also  such 
expressions  as  a  books,  n  bicycles,  x  pounds,  y  men,  etc. 
Numbers  represented  by  letters  are  often  called  literal 
numbers. 

In  the  case  of  literal  numbers  the  operations  of  addition, 
subtraction,  etc.  may  be  indicated  just  as  these  operations 
are  indicated  with  arithmetical  numbers.    Thus,  for  example, 

1 


2  HIGH  SCHOOL  ALGEBRA  [Ch.  I 

if  n  represents  one  number  and  k  another,  then  n-{-k  stands 
for  their  sum,  n  —  k  for  their  difference,  n  x  k   for  their 

product,  n-i-k  or   -  for  their  quotient. 
k 

EXERCISE  I 

If  a  stands  for  3,  b  for  2,  and  x  for  12,  find  the  value  of  each 
of  the  following  expressions  : 

-     2  a  -\-  X                            o     3  abx  —  ah 
3.     •  o.     • 

h  ab-{-hx 

^    a-\-hx  ^    X       4:a       x 


1. 

a-\-h. 

2. 

x  —  a. 

3. 

x-i-a. 

4. 

5b--* 

3a  a       2b       2 

7     ah -{-2  X  —10  ^  10    ^  4-  ^~^^ 

4a-f  46  a  b 

11.  If  s  represents  16,  what  number  is  represented  by  2s? 

s  3  s 

by  Js,  or  (as  usually  written)  -?     by  —  ? 

4  8 

12.  If  a  suit  of  clothes  costs  8  times  as  much  as  a  hat,  and 
if  h  stands  for  the  cost  of  the  hat,  how  may  the  cost  of  the  suit 
be  represented  ? 

13.  Does  h-\-Sh  (i.e.,  9h)  represent  the  combined  cost  of  the 
hat  and  suit  in  Ex,  12  ?     Explain  your  answer. 

14.  The  side  of  a  square  is  5  feet  long.  How  long  is  the 
bounding  line  of  this  square?  How  long  is  the  bounding  line 
if  the  side  is  x  feet  long  ? 

15.  A  boy's  present  age  is  15  years ;  indicate,  without  perform- 
ing the  subtraction,  his  age  4  years  ago.  What  was  his  age  n 
years  ago  ?     What  will  it  be  y  years  hence  ? 

16.  At  5  cents  each  how  many  erasers  can  be  bought  for  15 
cents  ?     for  x  cents  ?     for  n  dollars  ? 

17.  What  number  multiplied  by  8  gives  the  product  40?  If 
8 a;  =  40,  what  is  the  value  of  x?  If  5y -\-2y  =  21,  what  is  the 
value  of  y  ? 

*  6b  means  5  times  b  ;  so  too  ab  means  a  times  b  ;  and  3  ax  means  the 
product  of  3,  a,  and  x. 


2-^  INTliODUCTION  3 

3.  One  advantage  of  literal  numbers.  The  following  ex- 
amples show  how  the  solution  of  problems  may  often  be 
simplified  by  using  letters  to  represent  numbers. 

Prob.  1.  A  gentleman  paid  $45  for  a  suit  of  clothes  and  a 
hat.  If  the  clothes  cost  8  times  as  much  as  the  hat,  what  was 
the  cost  of  each  ? 

ARITHMETICAL    SOLUTION 

The  hat  cost  "a  certain  sum/'  and  since  the  clothes  cost  8  times 
as  much  as  the  hat,  therefore  the  cost  of  the  clothes  was  8  times 
"  that  sum,"  and  the  cost  of  the  two  together  was  9  times  "  that 
sum."  Hence  9  times  "that  sum"  is  $45,  and  therefore  'Hhat 
sum"  is  $5,  and  8  times  "that  sum"  is  $40;  i.e.,  the  hat  cost 
$5,  and  the  clothes  $40. 

This  solution  may  be  put  into  the  following  more  systematic 
form,  still  retaining  its  arithmetical  character. 

A  certain  sum  =  the  cost  of  the  hat ; 
then  •.•  *      8  tim,es  that  sum  =  the  cost  of  the  clothes, 

9  times  that  sum  =  the  cost  of  both, 
i.e.j  9  times  that  sum  =  $45. 

that  sum  =  $  5,  the  cost  of  the  hat, 
and  8  times  that  sum  =  $  40,  the  cost  of  the  clothes. 

ALGEBRAIC    SOLUTION 

The  solution  just  given  becomes  much  simpler  if  we  let  a 
single  letter,  say  x,  stand  for  the  number  of  dollars  in  "  a  certain 
sum  "  and  "  that  sum  "  as  used  above,  thus  : 

Let  X  =  the  number  of  dollars  the  hat  cost. 

Then  Sx  =  the  number  of  dollars  the  clothes  cost, 

and        x  +  S  x  =  the  number  of  dollars  both  cost ; 
i.e.,  9  a;  =  45. 

a;  =  5,  and  8  a?  =  40 ; 
i.e.,  the  hat  cost  $5,  and  the  clothes  cost  $40. 

N.B.     The  letter  x,  above,  stands  for  a  number,  not  for  the  cost  of  the  hat. 

*  The  symbols  •.•  and  .-.  stand  for  the  words  "shice"  and  "therefore," 
respectively. 


4  HIGH  SCHOOL  ALGEBRA  [Ch.  I 

Prob.  2.  If  a  locomotive  weighs  3  times  as  much  as  a  car,  and 
the  difference  between  their  weights  is  50  tons,  what  is  the  weight 
of  the  locomotive  ? 

SOLUTION 

Let       w  =  the  number  of  tons  in  the  weight  of  the  car. 
Then     Sw  =  the  number  of  tons  in  the  weight  of  the  locomotive, 
and,  since  the  difference  between  their  weights  is  50  tons, 

3w  —  w  =  50, 
i.e.f  2w  =  50, 

whence  iv  =  25, 

and  3w  =  75; 

i.e.,  the  locomotive  weighs  75  tons. 

Prob.  3.  Of  three  numbers  the  second  is  5  times  the  first,  and 
the  third  2  times  the  first ;  if  the  sum  of  these  numbers  equals 
the  third  number  increased  by  42,  what  are  the  numbers  ? 

SOLUTION 

Let  n  =  the  first  of  the  three  numbers. 

Then  5n  =  the  second  number, 

and  2  71=  the  third  number ; 

now  since  the  sum  of  the  three  equals  the  third  number  increased 
by  42, 

.-.  n-{-5n-j-2n  =  2n  +  4:2, 
i.e.,  Sn  =  2n-\-  42, 

hence  6  n  =  42.     [Subtracting  2  n  from  each  of  the  equal 

sums  above.] 
n  =  7,     5n  =  35,  and  2  n  =  14 ; 
i.e.,  the  numbers  are  7,  35,  and  14,  respectively. 

Remark.  Observe  that  the  steps  in  each  of  the  foregoing  solu- 
tions are : 

1.  To  let  some  letter,  say  x,  stand  for  one  of  the  unknown 
numbers  (preferably  the  smallest). 

2.  To  express  the  other  unknown  numbers  in  terms  of  x. 

3.  To  translate  into  algebraic  language  those  relations  between 
the  unknown  numbers  which  the  problem  states  in  words ;  this 


3]  INTRODUCTION  5 

translation  gives  an  equation,  and  from  it  the  required  numbers 
are  easily  found. 

Observe  also  that  while  the  above  problems  can  be  solved  by 
arithmetic,  the  algebraic  solution  is  much  simpler. 

EXERCISE  11 

Solve  the  following  problems : 

4.  In  a  room  containing  45  pupils  there  are  twice  as  many 
boys  as  girls.     How  many  boys  are  there  in  the  room  ? 

5.  If  a  horse  costs  7  times  as  much  as  a  saddle,  and  if  the 
difference  in  the  cost  of  the  two  is  $  90,  find  the  cost  of  each. 

6.  A  house  is  worth  5  times  as  much  as  the  lot  on  which  it 
stands,  and  the  two  together  are  valued  at  $4200.  Find  the 
value  of  each. 

7.  If  the  house  and  lot  of  Ex.  6.  differ  in  value  by  $4200,  how 
much  is  each  worth  ? 

8.  The  double  of  a  certain  number  taken  from  10  times  the 
same  number  leaves  72.     What  is  the  number  ? 

9.  If  n  represents  a  certain  number,  how  may  we  represent: 

(1)  the  number,  plus  4  times  itself,  plus  5  times  itself  ? 

(2)  the  sum  of  the  number,  its  double,  and  its  half  ? 
What  does  5  -a? -f-  7  n  —  3  n  represent  ? 

10.  A  number,  plus  twice  itself,  plus  4  times  itself,  is  equal  to 
56.     What  is  the  number  ? 

11.  Divide  98  into  three  parts  such  that  the  second  is  twice 
the  first  and  the  third  is  twice  the  second. 

12.  Divide  160  into  three  parts  such  that  two  of  them  are  equal, 
while  the  third  is  twice  either  of  the  others. 

13.  In  a  yachting  party  consisting  of  36  persons,  the  number 
of  children  is  3  times  the  number  of  men,  and  the  number  of 
women  is  one  half  that  of  the  men  and  children  combined.  How 
many  women  are  there  in  the  party  ? 

14.  If  I  have  s  nickels,  how  many  cents  have  I  ?  How  many 
cents  in  s  dimes?  in  s  quarters?  in  the  sum  of  s  nickels,  s 
dimes,  and  s  quarters  ? 


6  TUGn  SCHOOL  ALGEUnA  [Ch.  I 

15.  A  boy  found  that  he  had  the  same  number  of  5,  10,  and  25 
cent  pieces,  and  that  the  total  amount  of'  his  money  was  ^  3.20. 
How  many  coins  of  each  kind  had  he  ? 

16.  Alice  buys  Christmas  gifts  at  25  cents,  15  cents,  and  10 
cents  —  the  same  number  at  each  price.  If  she  spends  $2  in 
all,  how  many  gifts  does  she  buy  ? 

17.  If  a;  stands  for  a  certain  number,  what  would  stand  for 

(1)  the  double  of  the  number,  increased  by  7  ? 

(2)  the  difference  between  3  times  the  number  and  8  ? 

18.  How  would  you  represent  two  numbers  whose  difference  is 
4  ?    two  numbers  whose  sum  is  13  ? 

19.  Find  the  number  whose  double,  with  4  added,  equals  46. 

20.  Find  two  numbers,  differing  by  7,  whose  sum  is  35.  Also 
find  two  numbers  whose  sum  is  Q^  and  whose  difference  is  15. 

21.  William  has  8  cents  more  than  his  sister  Harriet,  and 
together  the  two  have  80  cents.  How  much  money  has  each? 
If  Harriet's  money  is  made  up  of  an  equal  number  of  nickels  and 
one-cent  pieces,  how  many  nickels  has  she  ? 

22.  In  a  family  of  seven  children  each  child  is  2  years  older 
than  the  next  younger.  If  the  sum  of  their  ages  is  84  years,  how 
old  is  the  youngest  child  ? 

23.  A  father's  age  is  now  3  times  that  of  his  son  ;  5  years  hence, 
the  sum  of  their  ages  will  be  62  years.  Find  the  present  age  of 
each.     (Cf.  Ex.  15,  p.  2.) 

24.  Four  years  ago  Isabel  was  twice  as  old  as  Mabel ;  the  sum 
of  their  present  ages  is  32  years.     How  old  is  each  ?    • 

25.  In  a  business  enterprise,  the  combined  capital  of  A,  B,  and 
C  is  f  21,000.  A's  capital  is  twice  B's,  and  B's  is  twice  C's. 
What  is  the  capital  of  each  ? 

26.  In  the  triangle  MNP,  NP'is  2  inches  longer 
than  MN,  while  PM  and  JfJVare  of  equal  length. 
If  the  sum  of  the  three  sides  is  S6  inches,  find 
the  length  of  each. 

27.  An   east-bound   and    a  west-bound   train 
leave  Chicago  at  the  same  hour,  the  first  running  twice  as  fast 


N  P 


3]  INTRODUCTION  7 

as  the  second ;  after  one  hour  they  are  90  miles  apart.     Find  the 
speed  of  each. 

28.  In  a  fishing  party  consisting  of  four  boys,  two  of  the  boys 
caught  each  the  same  number  of  fish,  another  caught  2  more  than 
this  number,  and  the  fourth,  1  less.  If  the  total  number  of  fish 
caught  was  29,  how  many  did  each  catch  ? 

29.  An  estate  valued  at  $24,780  is  to  be  divided  among  a  family 
consisting  of  a  mother,  two  sons,  and  three  daughters.  If  the 
daughters  are  to  receive  equal  shares,  each  son  twice  as  much  as  a 
daughter,  and  the  mother  twice  as  much  as  all  the  children  to- 
gether, what  will  be  the  share  of  each  ?  ^  ^ 

30.  ABCD  represents  the  floor  of  a  room. 
Find  the  dimensions  of  the  floor  if  its  bound- 
ing line  is  48  feet  long. 

31.  A  gallon  of  cream  is  poured  into  two 


(a;+4)  feet 


pitchers,   one    of   which   holds    7    times  as    -D  G 

much  as  the  other.     How  many  gills   does  each  pitcher  hold? 

32.  If  i  of  a  number  is  added  to  the  number,  the  sum  is  120. 
What  is  the  number  ? 

Suggestion.    Let  3  x  =  the  number. 

33.  If  ^  of  a  number  is  added  to  twice  the  number,  the  sum  is 
35.     What  is  the  number  ? 

34.  Of  two  numbers,  twice  the  first  is  7  times  the  second,  and 
their  difference  is  75.     Find  the  numbers. 

Suggestion,     Let  1  x  =  the  first  number,  then  2  x  =  the  second. 

35.  An  estate  of  f  19,600  was  so  divided  between  two  heirs 
that  5  times  what  one  received  was  equal  to  9  times  what  the 
other  received.     What  was  the  share  of  each  ? 

36.  A  tree  whose  height  was  150  feet  was  broken  off  by  the 
wind,  and  it  is  found  that  3  times  the  length  of  the  part  left 
standing  is  the  same  as  7  times  that  of  the  part  broken  off.  How 
long  is  each  part  ? 

37.  If  two  boys  together  solved  65  problems,  and  if  8  times  the 
number  solved  by  the  first  boy  equals  5  times  the  number  solved 
by  the  second  boy,  how  many  did  each  boy  solve  ? 


8  HIGH  SCHOOL  ALGEBRA  [Ch.  I 

II.    ELEMENTARY   OPERATIONS 

4.  Addition.  In  algebra,  as  in  arithmetic,  such  an  ex- 
pression as  7  +  3  is  read  "  7  plus  3,"  and  means  that  3  is  to 
be  added  to  7. 

To  perform  this  addition  we  begin  at  7  and  count  3  for- 
ward, obtaining  the  result  10,  which  is  called  the  sum  of 
these  two  numbers.* 

So  also  if  a  and  b  stand  for  any  two  numbers  whatever, 
the  expression  a-\-b  is  read  "a  plus  5,"  and  means  that  h  is 
to  be  added  to  a. 

The  result  obtained  by  adding  two  or  more  numbers  is 
called  their  sum,  and  the  numbers  that  are  to  be  added  are 
called  the  summands. 

5.  Subtraction.  To  what  number  must  3  be  added  to 
obtain  the  sum  8?  If  8  was  obtained  by  adding  3  to  some 
number  (i.e.^  by  counting  3  forward},  how  may  we,  starting 
with  8,  find  the  number  at  which  the  counting  began  ? 

Here,  as  in  arithmetic,  the  operation  of  finding  this  num- 
ber is  indicated  by  the  expression  8  —  3,  which  is  read  "  8 
minus  3."     We  may  say  that  8  —  3=5  because  5  +  3  =  8. 

The  process  of  finding  one  of  two  numbers  when  their 
sum  and  the  other  number  are  given,  is  called  subtraction. 
It  consists,  as  we  have  just  seen,  in  counting  backward,  i.e., 
in  undoing  the  work  of  addition,  which  consists  in  counting 
forward. 

If  a  and  b  stand  for  any  two  numbers  whatever,  the  ex- 
pression a  —  b  is  read  "  a  minus  5,"  and  means  that  b  is  to  be 
subtracted  from  a. 

The  result  obtained  by  subtracting  one  number  from  an- 
other is  called  their  difference  (also  the  remainder).  The 
number  which  is  to  be  subtracted  is  called  the  subtrahend,  and 

*  If  fractions  are  to  be  added,  we  first  reduce  tliera  to  a  common  denomi- 
nator and  tlien  add  their  numerators  ;  it  is  still  a  counting  process. 


4-C]  INTRODUCTION  9 

the  one   from  which  the  subtraction  is  to  be  made  is  called 
the  minuend. 

6.  Inverse  operations.  Of  two  operations  which  neutralize 
each  other  when  performed  in  succession,  each  is  called  the 
inverse  of  the  other.  Thus  the  operations  of  addition  and 
subtraction  are  each  the  inverse  of  the  other  (cf.  Exs. 
8-10,  below). 

EXERCISE- III 

Read  each  of  the  following  expressions,  then  name  its  parts : 
1.    8  +  12  =  20.  2.   9-7  =  2.  3.   12y-9y  =  3y, 

4.  Since  4  +  9  =  13,  therefore  13  -  9=  ?     13  -4  =  ? 

5.  In  subtraction,  what  name  is  used  to  denote  the  given  sum  ? 
the  given  summand?  the  required  summand  ?  Illustrate,  using 
Ex.  2  above. 

6.  Add  4  to  7  by  counting.  Where  do  you  begin  to  count  ? 
In  what  direction  do  you  count? 

7.  By  counting,  subtract  4  from  11.  Do  you  count  in  the 
same  direction  as  in  Ex.  6  ? 

8.  How  may  you  combine  the  subtrahend*  and  remainder  to 
get  the  minuend  ?     Why  ? 

9.  How  would  you  test  the  correctness  of  an  answer  in  sub- 
traction? Illustrate.  Could  you  use  subtraction  to  test  the 
correctness  of  a  sum  ? 

10.  When  is  one  operation  said  to  be  the  inverse  of  another  ? 
Using  the  numbers  8  and  6,  illustrate  the  fact  that  subtraction  is 
the  inverse  of  addition. 

11.  If  m  and  n  stand  for  any  two  given  integers  whatever,  can 
you,  by  counting,  find  the  value  of  ?7i  +  n  ?     of  m  —  n  ? 

12.  What  is  the  value  of  5  -  3  ?  of5-4?  of5-5?  of5-6? 
of  5  —  8  ?  In  order  that  subtraction  be  possible,  how  must  the 
subtrahend  compare  in  size  with  the  minuend  ?"* 

*  With  our  present  (arithmetical)  meaning  of  number  such  a  subtraction 
as  5  —  8  is,  of  course,  impossible ;  in  Chapter  II,  however,  we  shall  so  extend 
the  meaning  of  number  as  to  make  the  subtraction  a  —  b  possible  even  when 
b  is  greater  than  a. 


10  niGH  SCHOOL  ALGEBRA  [Cit.  I 

7.  Multiplication,  (i)  In  arithmetic,  multiplication  is 
usually  defined  as  the  process  of  taking  (additively)  one 
of  two  numbers,  called  the  multiplicand,  as  many  times  as 
there  are  units  in  the  other,  called  the  multiplier.  In  this 
sense,  6x4  (read  "6  multiplied  by  4")  means  6  +  64-6  +  6; 
i.e.^  this  multiplication  may  be  regarded  as  an  abbreviated 
addition. 

Strictly  speaking,  however,  the  above  definition  of  multi- 
plication applies  only  when  the  multiplier  is  an  arithmetical 
integer :  under  this  definition,  for  instance,  we  could  not  find 
such  a  product  as  8  x  51,  because  we  could  not  take  the  mul- 
tiplicand two  thirds  of  a  time  any  more  than  we  could  fire  a  gun 
two  thirds  of  a  time. 

(ii)  A  broader  definition  of  multiplication,  and  one  bet- 
ter suited  to  our  present  purpose,  may  be  stated  thus  : 

Multiplication  is  the  process  of  performing  upon  one  of 
two  given  numbers  (the  multiplicand)  the  same  operation 
as  that  which  is  performed  upon  unity  to  get  the  other  given 
number  (the  multiplier)  ;  the  result  thus  obtained  is  called 
the  product  of  these  numbers.  The  multiplicand  and  multi- 
plier are  called  factors  of  the  product. 

To  illustrate,  consider  again  the  question  of  multiplying 
8  by  5|-.  The  multiplier,  5|-,  is  obtained  from  unity  by  tak- 
ing the  unit  5  times,  and  J  of  the  unit  twice,  as  summands, 
i.e.,  5f  =  1  +  1  +  1  +  1  +  1  +  1+ J; 

and,  therefore,  by  this  new  definition  of  multiplication, 
8x5f  =  8  +  8  +  8  +  8  +  8  +  |  +  f  =  40  +  J^  =  45f 

(iii)  Just  as  6  X  5  means  that  6  is  to  be  multiplied  by  5, 
so  5  X  3  means  that  b  is  to  be  multiplied  by  3.  Similarly, 
kxn  xy  means  that  h  is  to  be  multiplied  by  n^  and  that 
their  product  is  then  to  be  multiplied  by  y. 

Instead  of  the  oblique  cross  (  x  ),  a  center  point  (•)  placed 
between  two  numbers  (a  little  above  the  line  to  distinguish 
it  from  the  decimal  point)  is  frequently  used  as  a  sign  of 


7-8]  INTttOhVCTlON  .       11 

multiplication.  And  even  the  center  point  is  usually  omitted 
if  doing  so  causes  no  confusion.  Thus,  8xri  =  o-'/i=3n; 
so,  too,  p  xr  xt=p  •  r  '  t=prt,  and  3  x  7  =  o  •  7.  But  the 
sign  (cross  or  center  point)  must  not  be  omitted  between  two 
arithmetical  numbers.     (Why  not?) 

EXERCISE  IV 
Kead  each  of  the  following  expressions,  then  name  its  parts : 
1.   8x3  =  24.  2.   f.  15  =  10.  3.   5«.4  =  20a. 

4.  What  is  the  value  of  5  •  3  ?  How  is  this  product  obtained 
under  the  old  definition  of  multiplication  [§  7  (i)]  ? 

5.  Using  the  new  definition  [§  7  (ii)],  show  that  5  •  3  means 
5  +  5  +  5.     Similarly,  explain  the  meaning  of  9  •  4;  of  4  •  9. 

6.  Show  that  2|  •  8  has  the  same  meaning  under  the  old  defi- 
nition of  multiplication  as  under  the  new. 

7.  To  get  f  from  1,  we  divide  1  into  how  many  equal  parts  ? 
How  many  of  these  parts  do  we  take '?  What,  then,  should  be 
done  to  10  in  multiplying  it  by  J  ? 

As  in  Ex.  7,  find  the  following  products : 

8.  16.  f.  9.    12.  2f  10.   7.5f. 

If  a  =  2,  6  =  5,  and  a?  =  |,  find  the  value  of  each  of  the  follow- 
ing expressions : 

11.  Sabx.  13.   2bx  —  ax  +  3b.  15.   Saax  +  4:bx. 

12.  5b  +  6x-ab.        14.    7abx  +  3a-2b.        16.    aab  — 10 x. 

8.  Division.  Division  is  the  inverse  (i.e.^ihe  "undoing") 
of  multiplication.  Thus,  since  4  x  9  =  36,  therefore  36  -=-  9  =  4, 
and  36 -4  =9. 

The  expression  36  -^  9  =  4  is  read  "  36  divided  by  9  equals 
4."  Here  36  is  called  the  dividend,  9  the  divisor,  and  4  the 
quotient. 

In  multiplication  we  have  given  two  numbers,  and  are 
asked  to  find  their  product ;  in  division  we  have  given  the 
product  (now  called  the  dividend}  and  one  of  the   factors 

HIGH  SCH.  ALG.  —  2 


1^  tilGH  SCHOOL  ALGEBRA  [Ch.  1 

(now  called  the  divisor')^  and  are  asked  to  find  the  other 
factor  (now  called  the  quotient). 

Hence  we  may  say :  division  is  the  process  of  finding  from 
two  given  numbers,  called  dividend  and  divisor,  respectively, 
a  third  number  (called  the  quotient)  such  that  the  divisor 
multiplied  by  the  quotient  equals  the  dividend. 

E.g.,  36  --  9  =  4,  because  4  x  9  =  36. 

If  8  and  t  represent  any  two  numbers  whatever,  then  each 

of  the  expressions,  s  -?-  f ,  -,  s/U  and  s  :  t  indicates  that  s  is  to 
be  divided  by  t. 

If  the  divisor  is  not  exactly  contained  in  the  dividend, 
then,  as  in  arithmetic,  the  indicated  division  is  called  a 
fraction. 

E.g.^  ^,  -— ,  — ,  and  — ^t_  are  called  fractions. 
6     D     n  y  . 

It  is  to  be  remarked,  however,  that  literal  numbers 
may   be    fractional    in   form    but    integral    in    value,    and 

vice  versa.  Thus,  -,  though  fractional  in  form,  has  the  inte- 
gral value  3  if  a  =  12  and  5  =  4. 

9.  Powers,  exponents,  etc.  (i)  In  algebra,  as  in  arithmetic, 
such  a  product  as  5  •  5  •  5  is  usually  written  in  the  abbrevi- 
ated form  5^,  the  small  3  showing  the  number  of  times  that 
5  is  used  as  a  factor. 

Similarly,  23  z=  2  •  2  •  2,  a^^a-a-a,  2^  .  52=  2  •  2  •  2  •  5  •  5, 
7i2jt>4  z=  n^ '  p^  =  n  '  71  •  p  '  p  '  p  •  p,  etc. 

The  expression  k^  is  usually  called  the  fourth  power  of  h. 
In  this  expression,  4  is  called  the  exponent  and  h  the  hase  of 
the  power. 

(ii)  Hence  the  following  definitions  :  A  power  of  a  number 
is  the  product  arising  from  using  the  given  number  one  or 
more  times  as  a  factor. 

An  exponent  is  a  number  placed  (in  small  symbols)  at  the 


8-9]  INTRODUCTION  "  13 

right  and  slightly  above  a  given  number,  to  show  how  many 
times  the  latter  is  to  be  used  as  a  factor. 

Thus,  if  X  represents  any  number  whatever,  and  n  any 
arithmetical  integer,*  then  the  expression  x^  is  called  the  nth 
power  of  07,  and  means  the  product  arising  from  using  x  as 
a  factor  n  times ;  n  is  the  exponent  of  the  power. 

Note.  Observe  that  under  the  above  definitions  a^  has  the  same  mean- 
ing as  a  ;  the  exponent  1,  therefore,  need  not  be  written. 

The  second  and  third  powers  of  numbers  are,  for  geometrical  reasons, 
often  called  by  the  special  names  of  square  and  cube  respectively.  Thus  a^ 
is  called  "  the  second  power  of  a,"  "  the  square  of  a,"  and  also  "  a  squared." 

EXERCISE  V 

Eead  each  of  the  following  expressions,  name  its  parts,  and 
test  the  correctness  of  the  results : 

1.  18-6  =  3.  4.  ?5^  =  4a.  7.   ^  =  Z2. 

2.  28^14  =  2.  5.   9.^81.  k^^4a^ 

3.  6_|o  =  70.  6.   2^.32  =  288.  *      32  3  * 
Read  the  following  expressions  and  tell  what  operations  are 

indicated  in  each  case;  then  find  the  numerical  value  of  each 
expression  when  a  =  5,  6  =  2,  7i  =  l,  and  ic  =  4. 

.5  ,^    3  6^  „     a^-lOW 


9.    a\  12.   ^^1^.  15. 

16  3  a 

10-   «'  +  ^'-  13.   8W-10a^.       ^^     75 -aW 

11.   7wV.  14.   7i^  +  bax\  '        y?-a    ' 

17.  Write  7«7-7«7  by  means  of  the  exponent  notation. 
Also  a  '  a'a\  5  •  5  •  a;  •  a;  •  a; ;  and  9  •  9  •  9  •  9  •  a  •  a  •  ?/  •  ?/  •  2/. 

18.  How  may  we  use  multiplication  to  test  the  correctness  of 
an  example  in  division  ?     Why  ? 

19.  The  sum  of  any  two  integers  is  integral.  Is  this  true  of 
their  difference  ?  of  their  product  ?  of  their  quotient  ?  Illus- 
trate your  answers. 

*  We  shall  later  (Chapter  XVI)  enlarge  the  scope  of  such  a  symbol  as  x"  by 
giving  it  a  meaning  even  when  n  does  not  represent  an  arithmetical  integer. 


14  HIGH  SCHOOL  ALGEBRA  [Ch.  I 

20.  When  the  dividend  is  not  exactly  divisible  by  the  divisor, 
what  name  is  given  to  the  indicated  quotient  ? 

5 

21.  How   are    fractions   defined   in  arithmetic  ?     Is  —  a  frac- 

tion  under  the  arithmetical  definition  ?     If  not,  why  not  ? 

10.  The  order  in  which  arithmetical  operations  are  to  be 
performed.  What  is  the  value  of  2  +  6  •  5  —  8  ^  2  ?  Is  it 
28,  16, 'or  12?  In  order  that  such  an  expression  shall  have 
the  same  meaning  for  all  of  us,  mathematicians  have  agreed 
that,  when  there  is  no  express  statement  to  the  contrary : 

(1)  A  succession  of  multiplications  and  divisions  shall 
mean  that  these  operations  are  to  be  performed  in  the  order 
in  which  they  occur  from  left  to  right. 

(2)  A  succession  of  additions  and  subtractions  shall  mean 
that  they  are  to  be  performed  in  the  order  in  which  they 
occur. 

E.g.,  9  .  8  --  6  . 2  =  72  --  6  .  2  =  12  .  2  =  24, 

but  9  •  8  -J-  6  •  2  is  710^  equal  to  72  -h  12,  i.e.,  to  6. 

So,  too,       7  +  9-6  +  3  =  16 -6  +  3  =  10 +  3  =  13, 
but  7  +  9  —  6  +  3  is  710^  equal  to  16  —  9,  i.e.,  to  7. 

(3)  A  succession  of  the  operations  of  addition,  subtrac- 
tion, multiplication,  and  division  shall  mean  that  all  the 
operations  of  multiplication  and  division  are  to  be  performed 
before  ani/  of  those  of  addition  and  subtraction,  and  in  accord 
with  (1)  above.  The  additions  and  subtractions  are  then 
to  be  performed  in  accord  with  (2)  above. 

E.g.,  2  +  6- 5-8--2  =  2  + 30 -4  =  28. 

Note.  While  such  an  expression  as  3  •  a  -^  2  •  a;  •  ?/  means  [(8  a)  ^  2]  •  a;  •  y, 
the  expression  Sa  -^2xy  is  usually  understood  to  mean  (3  a)  ~  (2  xy)  ; 
i.e.,  3  a  and  2  xy  are  here  understood  to  vei>resent products  rather  than  unper- 
formed multiplications. 

11.  Signs  of  aggregation,  (i)  Any  desired  departure  from 
the  order  of  operations  given  in  §  10  may  be  indicated  by 
employing  one  or  more  of  the  so-called  signs  of  aggregation ; 


1-11]  tNTROTWCrtON  15 

among  these  are  the  parenthesis  (  ),  the  brace  \\,  the  bracket 
[  ],  and  the  vinculum  . 

(ii)  An  expression  within  a  parentliesis,  brace,  or  bracket, 
or  under  a  vinculum,  is  to  bo  regarded  as  a  whole,  and  is  to 
be  treated  as  though  it  were  represented  by  a  single  symbol. 

U.g.,  (2  +  6)  .  5  -  3  -  (7  +  8  -^  2)  =  8  •  5  ^  3  -  11,  i.e,, 
'2^.  So,  too,  (4  +  6)  -^  2  =  5,  while  without  the  parenthesis 
its  value  would  be  7. 

It  may  sometimes  be  useful  even  to  employ  one  sign  of 
aggregation  within  another, 

Mg.,  72- J47-7(15-10)S  =72- ^47-35?  =72-12  =  6. 

EXERCISE  VI 
Find  the  value  of  each  of  the  following  expressions: 

1.  20  +  5-3.  4.   12-2x4.  7.   16-2-1-6. 

2.  20-5  +  3.  5.    12 -(2x4).  8.   16  -  (2  +  6). 

3.  20 -(5 +  3).  6.   9.(6-2).  9.   11  •  4  -  6  •  3-2. 

10.  28 -(6 +  13) -(10 -2).         13.   42-7x5-5  +  6x2. 

11.  32-9  +  6--2  +  1.  14.   12  +  9-3-30-2  +  8. 

12.  32- (9  +  6) --(2  +  1).  15.    (12-f  9).3-(30-2  +  8). 

16.    J25 -  (10 +  13)S -2  +  31-5  +  4. 


17.   16- 9 -4(36 -3-2) +54 -(17 -12-5). 
Read  each  of  the  following  expressions,  and  tell  in  what  order 
the  indicated  operations  are  to  be  performed  : 

18.  ac-^b.  21.    (c-by.  24.    6aH-2c-2d^ 

19.  a(c-^b\  22.    c'-^b'-2d.  

20.  c-b\  23.    c----(b^-2d).         25.    ^  -— . 

^  ^  c<i^+[2(c  — a)]- 

26.    If  a  =  8,  b  —  3,  c  =  12,  and  2d=  1,  find  the  numerical  value 
of  each  of  the  expressions  in  Exs.  18-25. 


CHAPTER   II 

POSITIVE    AND    NEGATIVE    NUMBERS 

12.  Introductory.  *  Suppose  the  present  reading  of  a  ther- 
mometer is  5°  above  zero,  and  the  temperature  is  falling ; 
what  will  be  the  reading  when  it  has  fallen  1°?  2°?  3°? 
4°?     5°?     6°?     7°?     8°? 

*  Note  to  the  Teacher.  It  will  stimulate  interest  in  the  work,  as  well 
as  enlarge  the  pupil's  view,  if  the  teacher  will  amplify  and  present  to  the 
class  the  following  considerations  (cf.  also  El.  Alg.  pp.  18,  19)  : 

1.  Man's  earliest  idea  of  number  came  from  counting.,  and  in  this  way 
there  arose  the  number  system  consisting  of  the  integers  1,  2,  3,  4,  •••. 
Presently  he  advanced  a  step  and  counted  backward  as  well  as  forward,  and 
by  groups  of  things  as  well  as  by  single  things ;  this  led  the  way  to  the  opera- 
tions of  addition,  subtraction,  multiplication,  and  division. 

2.  Having  devised  this  number  system  and  these  operations  to  meet  the 
needs  of  his  daily  life  (just  as  later  he  invented  the  clock,  the  steam  engine, 
the  telephone,  etc.),  he  soon  found  the  system  inadequate:  7  h- 3,  for  ex- 
ample, represents  no  number  in  the  above  system.  His  increasing  desire  for 
exactness,  however,  as  civilization  advanced,  demanded  that  division  should 
be  always  possible;  for  this  and  other  reasons  he  invented  fractions,  and 
included  them  in  his  number  system . 

3.  Many  of  the  things  with  which  we  are  concerned  bear  a  relation  of 
opposition  to  each  other  (assets  and  liabilities,  thermometer  readings  above 
zero  and  below  zero,  etc.),  and  the  change  from  one  of  these  opposites  to 
the  other  may  be  regarded  as  a  subtraction  (cf.  §  12)  ;  for  this  and  other 
reasons  it  was  found  advantageous  again  to  extend  the  number  system,  and 
thus  to  make  subtraction  always  possible. 

4.  The  use  of  literal  notation,  too,  quite  apart  from  the  foregoing  con- 
siderations, demands  a  number  system  in  which  addition,  subtraction,  etc., 
are  always  possible.  If  subtraction,  for  example,  is  not  always  possible, 
such  an  expression  as  a  —  6  may  or  may  not  represent  a  number ;  hence, 
should  a  —  h  occur  in  the  solution  of  a  problem  before  the  relative  values  of 
a  and  b  were  known,  our  work  would  come  to  a  standstill.  It  is  wiser,  there- 
fore, to  include  negatives  in-  our  number  system,  and  let  the  solution  proceed, 
giving  the  necessary  interpretation  to  our  results  later. 

16 


12-18]  POSITIVE  AND  NEGATIVE  NUMBERS  17 

Wliicli  one  of  the  arithmetical  operations  (addition,  sub- 
traction, etc.)  did  you  use  in  answering  these  questions? 

Again,  if  a  business  man's  financial  losses  are  large  enough, 
they  will  decrease  his  capital,  not  only  to  zero  but  through 
zero,  and  bring  him  into  debt. 

So,  too,  if  a  traveler  now  in  north  latitude  goes  south  far 
enough,  he  will  pass  through  zero  latitude,  and  into  south 
latitude. 

Observe  that,  in  each  of  the  statements  just  made,  the 
change  from  a  given  condition  to  its  opjjosite  is  essentially 
a  process  of  subtraction;  it  is  a  subtraction,  moreover,  in 
which  we  can  subtract  not  only  to,  but  through  zero. 

In  our  present  number  system,  we  can  subtract  only  to 
(not  through^  zero ;  in  order,  therefore,  to  express  m  the 
simplest  possible  wag  the  numerical  relations  between  such 
opposite  things  as  those  given  above  (gains  and  losses,  lati- 
tude north  and  south,  etc.),  we  must  extend  our  number 
system  so  as  to  make  subtraction  through  zero  possible. 

13.   The  number  system  extended.     The  arithmetical  inte- 
gers arranged  in  a  series  increasing  by  one  from  left  to  right, 
and  therefore  decreasing  by  one  from  right  to  left,  are 
1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  ... 

Addition  is  performed  by  counting  toward  the  right  (cf. 
§  4),  and  subtraction  by  counting  toward  the  left,  in  this 
series.  Moreover,  addition  is  always  possible  because  this 
series  extends  without  end  toward  the  right,  and  subtraction 
is  arithmetically  possible  only  when  the  subtrahend  is  not 
greater  than  the  minuend  because  this  series  is  limited  at 
the  left. 

Hence,  to  make  subtraction  with  arithmetical  integers 
always  possible,  it  is  only  necessary  to  continue  the  above 
series  indefinitely  toward  the  left. 

Let  the  result  of  subtracting  1  from  1  be  designated  by  0 ; 
of  subtracting  1  from  0,  by  —  1 ;   of  subtracting  1  from  —  1, 


18  HIGH  SCHOOL  ALGEBRA  [Ch.  II 

by  —  2 ;  of  subtracting  1  from  —  2,  by  —  3,  etc.  ;  with  these 
new  numbers  included,  the  series  becomes 

...,  -6,-5,-4,-3,-2,-1,  0,  1,  2,  3,  4,  5,  6,  7,  ..., 

whicli  extends  without  end  toward  the  left  as  well  as  toward 
the  right. 

Since  in  this  enlarged  series  each  number  is  less  by  one 
than  the  next  number  at  its  right  (and  hence  greater  by 
one  than  the  next  number  at  its  left),  therefore  addition  and 
subtraction  with  arithmetical  integers  may,  as  before,  be 
performed  by  counting  toward  the  right  and  left  respectively. 

E.g.^  to  subtract  8  from  5,  ^.e.,  to  find  the  number  which 
is  8  less  than  5,  we  begin  at  5  and  count  8  toward  the  left, 
arriving  at  —  3 ;  hence,  5  —  8  =  —  3. 

Similarly,  4-6=— 2,  4-9=- 5,  11 -16  =-5,  etc.  ; 
hence,  besides  indicating  a  particular  place  in  the  enlarged 
number  series,  —  5  also  indicates  that  the  subtrahend  is  5 
greater  than  the  minuend. 

Again,  to  add  7  to  —4,  i.e.^  to  find  the  number  which  is 
7  greater  than  —  4,  we  begin  at  —  4  and  count  7  toward  the 
right,  arriving  at  3 ;  hence  —4  +  7  =  3. 

Note.  Such  an  expression  as  4  —  9  =  —  5  is,  of  course,  not  to  be  under- 
stood to  mean  that  9  actual  units  of  any  kind  can  be  subtracted  from  4  such 
units  ;  4  of  the  9  units  may  be  immediately  subtracted,  leaving  the  other  5 
units  to  be  subtracted  later  if  there  is  anything  from  which  to  subtract ;  in 
this  sense  the  number  —5  may  be  said  to  indicate  a  postponed  subtraction, 
and  thus  to  have  a  subtractive  quality ;  hence  the  appropriateness  of  attaching 
the  minus  sign  to  such  numbers. 

14.  Positive  and  negative  numbers.  Numbers  which  are 
less  than  zero  *  are  called  negative  numbers ;  while  numbers 
greater  than  zero  are,  for  distinction,  called  positive  numbers. 

In  writing  positive  and  negative  numbers,  the  latter  are 
always  preceded  by  the  minus  sign,  while  positive  numbers 
may  be  written  either  with  or  without  the  plus  sign.     Thus, 

*  "Less  than  zero"  in  the  sense  suggested  in  §  13, 


13-15]  POSITIVE  AND  NEGATIVE  NUMBEES  19 

—  5,  —2.3,  —48,  and   —  |  are  negative  numbers,  while  3, 
+  84,  and  1.7  are  positive  numbers. 

Such  a  number  as  —  2  is  read  either  as  negative  two  or 
as  minus  two,  while  +  6  is  read  as  positive  six  or  plus  six. 

15.  Algebraic  numbers,  etc.  Positive  and  negative  num- 
bers together  (including,  of  course,  fractions  as  well  as  inte- 
gers) are  often  called  algebraic  numbers,  wliile  positive 
numbers  alone  are  called  arithmetical  numbers. 

By  the  absolute  value  of  a  number  is  meant  merely  its  size 
without  regard  to  its  quality/ ;  thus  —  2  and  +  2  have  the 
same  absolute  value;  so  also  have  —  17.25  and  +  17.25. 

Two  numbers  which  have  the  same  absolute  value  but 
which  are  of  opposite  quality,  are  called  opposite  numbers ; 
such,  for  example,  are  5  and  —  5. 

Note.  The  signs  +  and  —  as  used  above  are  called  signs  of  quality. 
We  shall  continue,  however,  to  use  them  as  signs  of  operation  also  ;  hence 
it  is  sometimes  necessary,  in  order  to  avoid  possible  confusion  arising  from 
this  double  use,  to  inclose  a  number  with  its  quality  sign  in  a  parenthesis 
(cf.  Ex.  20,  p.  21). 

EXERCISE  VII 
By  counting  along  the  algebraic  number  series  given  in  §  13 : 

1.  Add  3  to  8.  5.    Subtract  3  from  8. 

2.  Add  7  to  —  3.  6.    Subtract  7  from  7. 

3.  Add  4  to  —  4.  7.    Subtract  7  from  4. 

4.  Add  5  to  —  12.  8.    Subtract  5  from  —  3. 

9.   If  temperature  above  zero  is  indicated  by  positive  numbers, 
how  may  temperature  below  zero  be  indicated  ? 

10.  Interpret  the  following  temperature  record  taken  from  a 
U.  S.  Weather  Bureau  report :  Albany,  -f-  8° ;  Bismarck  (S.D.), 
- 11° ;  Buffalo,  -  2° ;  Denver,  -  5° ;  and  Galveston,  +  34°.     . 

11.  In  the  above  record  how  much  warmer  is  it  at  Albany  than 
at  Bismarck  ?  at  Buffalo  than  at  Denver  ?  at  Buffalo  than  at 
Galveston  ?     Illustrate  your  answers  by  diagrams. 


20  HIGH  SCHOOL   ALGEBRA  [Ch.  II 

12.  The  total  value  of  a  man's  property  (his  assets)  is  $  15,860, 
and  his  total  indebtedness  (liabilities)  is  $1420.  What  is  the  net 
value  of  his  estate?  What  is  the  net  value  of  an  estate  of  which 
the  assets  are  a  dollars  and  the  liabilities  h  dollars  ? 

13.  If  h  exceeds  a  in  Ex.  12,  is  a  —  6  positive  or  negative  ? 
How  should  a  —  6  be  interpreted  in  this  case  ?  What  is  meant 
by  saying  that  the  net  value  of  an  estate  is  —  $  750  ? 

14.  If  assets  are  indicated  by  positive  numbers,  interpret  the 
financial  conditions  indicated  by  the  following  numbers :  $  783 ; 
-  $  2568 ;   -  $  374.20 ;  and  $  (856-1232). 

15.  Kegarding  longitude  west  of  Greenwich  as  positive,  indi- 
cate by  a  number  and  sign  that  a  place  is :  (1)  in  24°  east  longi- 
tude; (2)  on  the  meridian  of  Greenwich;  (3)  in  10°  west  longitude; 
(4)  in  15°  east  longitude. 

16.  At  6  A.M.  a  thermometer  records  15°  below  zero;  by  noon 
it  has  risen  32°.  Indicate  by  a  number  and  sign  the  reading  at  6 
A.M.  and  at  noon.     Illustrate  by  a  diagram. 

17.  If  on  the  line  X'OX  distances  to  the   ^^ ? ^ 

left  of  0  are  called  negative,  locate  the  point 

on  this  line  whose  distance  from  0  is :   +  .3  in. ;  —  .5  in. ;  +  .1  in. 

18.  If  positive  numbers  are  used  to  denote  assets,  gains  in  busi- 
ness, increase  of  any  kind,  temperature  above  zero,  easterly  motion, 
south  latitude,  west  longitude,  distance  down  stream,  etc.,  what  are 
the  corresponding  meanings  to  be  attached  to  negative  numbers  ? 

19.  An  ocean  steamer  is  in  12°  east  longitude.  If  east  longitude 
is  indicated  by  positive  numbers,  and  if  the  vessel  moves  west- 
ward through  7°  of  longitude  per  day,  indicate  by  a  number  and 
sign  the  longitude  of  the  vessel  4  days  hence ;  1|-  days  hence ;  2 
days  ago.     Illustrate  by  a  drawing. 

20.  What  number  must  be  added  to  — 12  to  make  the  sum  4  ? 
Which,  then,  is  the  greater,   —12  or  4  ?     How  much  greater? 

"Which  of  these  numbers  has  the  greater  absolute  value  ? 

• 

16.  Addition  of  algebraic  numbers.  As  we  have  already 
seen  (§§  4,  13),  to  add  a  positive  number  to  any  given  num- 
ber, we  begin  at  the  given  number  and  aoxmi  forivard. 


la-lC]  POSITIVE  AND  NEGATIVE  NUMBERS  21 

Moreover,  since  a  negative  number  represents  an  unper- 
formed subtraction  (§  13,  Note),  therefore  adding  a  negative 
number  means  performing  this  subtraction.  Hence,  to  add 
a  negative  number  to  any  given  number,  we  begin  at  the  given 
number  and  count  hachvard. 

E.g.^  to  add  —  6  to  54  we  begin  at  54  and  count  6  backward; 
i.e.,  54  +  (-  6)  =  54  -  6  =  48  (cf.  Exs.  19,  20,  below). 

Such  an  expression  as  11—4—8  +  3,  which,  by  what  is 
said  above,  equals  11  +  (—  4)  +  (—  8)  +  3,  is  usually  spoken 
of  as  an  algebraic  sum. 

EXERCISE  VIII 
Perform  the  following  additions,  and  explain  your  work : 
1.  2.  3.  4.  5.  6. 


9 


12  -8  9  14  -7 

8  8-3  -6  -3 


7. 

8. 

9. 

10. 

11. 

12. 

-4 

23 

-11 

-31 

7 

-9 

-12 

-15 

-26 

45 

5 

—  5 

13. 

14. 

15.' 

16. 

17. 

18. 

-3 

7 

-6 

11 

-22 

-72 

4 

5 

-8 

-4 

31 

^b 

-8 

-9 

—  2 

-9 

15 

-21 

19.  If  a  boy  weighing  54  lb.  were  weighed  while  holding  a 
toy  balloon  which  pulls  upward  with  a  force  of  6  lb.,  what 
would  be  the  combined  weight?  If  +54  lb.  represents  the 
weight  of  the  boy,  what  would  represent  the  iveight  of  the 
balloon  ? 

20.  In  Ex.  19  the  combined  weight  of  the  boy  and  the  balloon 
is  (+54)  +  (—  6)  lb. ;  hence  adding  the  negative  number  destroys 
part  of  the  positive  number ;  is  this  true  in  general  for  additions 
of  positive  and  negative  numbers  ?     Illustrate  your  answer. 


22  HIGH   SCHOOL   ALGEBRA  [Ch.  II 

21.  When  does  the  addition  of  a  negative  number  to  a  positive 
number  destroy  the  latter  wholly  ?  When  only  in  part  ?  Illus- 
trate your  answers  by  means  of  assets  and  liabilities  (cf.  Ex.  12, 
p.  20). 

22.  Ex.  21  suggests  a  useful  rule  for  algebraic  addition,  viz.  : 

(1)  To  add  two  numbers  with  like  signs,  find  the  sum  of  their 
absolute  (arithmetical)  values,  and  to  this  prefix  their  common 
sign. 

(2)  To  add  two  numbers  with  unlike  signs,  find  the  difference 
of  their  absolute  values,  and  to  this  prefix  the  sign  of  the  larger. 

Test  this  rule  in  the  examples  on  p.  21. 

23.  A  wheelman  after  riding  37  miles  westward  from  a  certain 
point  rides  back  12  miles.  If  distances  to  the  westward  are  indi- 
cated by  positive  numbers,  show  that  37  +  (—  12)  miles  indicates 
both  his  direction  and  his  distance  from  the  starting  point. 

24.  Indicate  by  a  sum  of  positive  and  negative  numbers  what 
temperature  is  now  registered  by  a  thermometer  which  stood  at 
54°  above  zero,  then  rose  2°,  later  fell  9°,  and  then  rose  2i°. 

25.  Eind  the  following  indicated  algebraic  sums :  18  +  (—  3)  + 
(_10)  +  2;    42+(-27)4-(-64);    _5 +  18  +  (-11)  +  23. 

26.  Is  algebraic  addition  sometimes  performed  by  arithmetical 
subtraction  ?  Is  it  so  when  the  two  summands  have  like  signs  ? 
when  they  have  unlike  signs  ?     Illustrate  your  answers. 

17.  Subtraction  of  algebraic  numbers.  We  already  know 
how  to  subtract  positive  numbers  (of.  §§  5,  13);  therefore 
we  now  need  to  consider  only  the  question  of  subtracting 
negative  numbers. 

Now,  since  subtraction  is  the  inverse  of  addition  (§  6) 
and  since  7  + (-8)  =  4  (§  16),  therefore  4 -(-3)=  7;  i.e., 
4-(-3)  =  4  +  3.  Similarly:  11  -  (- 5)  =  11  +  5;  -3- 
(-12)= -3  + 12;   etc. 

Hence,  subtracting  a  negative  number  from  any  given 
number  gives  the  same  result  as  adding  its  opposite  to  the 
given  number. 


10-17]             POSITIVE  AND  NEGATIVE  I^ UMBERS  23 

EXERCISE  IX 

Subtract  the  numbers  written  below,  each  from  the  one  above 
it,  giving  the  necessary  explanation  in  each  case: 

1.                           2.                           3.                             4.  5. 

12                       4                     -2                         8  -4 

8                       7                         5                     -3  -2 


6. 

7. 

8. 

9. 

10. 

-3 

11 

6 

-4 

-31 

-10 

-8 

-15 

-12 

-25 

11.  The  weight  of  a  boy  while  holding  a  toy  balloon,  which 
pulls  upward  with  a  force  of  6  lb.,  is  48  lb.  If  we  take  away 
(subtract)  the  balloon,  how  much  will  the  boy  weigh  ?  Show  that 
in  this  case  48  —  (—  6)  =  54. 

Supply  the  missing  numbers  in  the  following  equations : 

12.  •.•-5h-?  =  -2,    .-.  _2-(-5)  =  ? 

13.  ...  _54-?  =  -9,    ...  -9-(-5)  =  ? 

14.  |-(_|)  =  ?  16.    10-3-(-5)  =  ? 

15.  _l|-(-5f)  =  ?  17.   23-(-a)  +  (-3)  =  ? 

If  a  =  5,  b  =  —  6,  c  =  —  3,  and  d  =  2,  find  the  value  of  each  of 
the  following  algebraic  sums : 

18.  a-b  +  c  +  d.  21.    a^^b-(d^-c). 

19.  a-^c  —  (b  —  d).  22.    ad  —  b-{-(a^  —  b). 

20.  a—(b  +  G-d).  23.    2  a-^  -  c  —  3d^-\-b. 

24.  Using  positive  numbers  to  represent  assets,  illustrate  the 
fact  that  subtracting  a  negative  from  a  positive  number  increases 
the  latter  (cf.  Ex.  11). 

25.  Make  up  concrete  examples  (like  Ex.  11  or  Ex.  24)  to  illus- 
trate Exs.  4,  8,  and  3,  above. 

26.  Solve  Exs.  1,  2,  5,  and  9,  above,  by  counting  ;  and  explain 
in  each  case  why  you  count  in  one  direction  rather  than  in  the 
opposite  direction. 


24  HIGH   SCHOOL  ALGEBRA  [Ch.  II 

27.  A  rule  for  subtraction  is  often  stated  thus :  "  Reverse  the 
sign  of  the  subtrahend  and  proceed  as  in  addition."  Show  that 
this  rule  is  correct  when  the  subtrahend  is  a  negative  number. 

28.  Mt.  Washington  is  6290  feet  above  sea  level,  Pikes  Peak 
is  14,083  feet  above  sea  level,  and  a  place  near  Haarlem,  in  Holland, 
is  16|^  feet  below  sea  level.  By  subtraction  find  how  much  higher 
Pikes  Peak  is  than  Mt.  Washington ;  and  also  how  much  higher 
Mt.  Washington  is  than  the  place  near  Haarlem. 

29.  When  is  algebraic  subtraction  equivalent  to  arithmetical 
addition  (cf.  Ex.  2%,  p.  22)  ?     Illustrate  your  answer. 

30.  Write  the  following  algebraic  sums  so  that  they  shall  not 
contain  minus  as  a  sign  of  operation,  then  find  the  value  of  each : 
36-19-13+2;    14a-26i«+9}a-15a;    -6a;-47a;-3a;. 

18.  Product  of  algebraic  numbers.  Rule  of  signs.*  The 
product  of  any  two  algebraic  numbers  is  tlie  result  obtained 
by  performing  upon  the  multiplicand  the  same  operation  as 
that  which  is  performed  upon  positive  unity  to  obtain  the 
multiplier  [§  7  (ii)]. 

E.g,,  since  3  =  1  +  1  +  1, 

therefore  8  •  3  =  8  +  8  +  8  =  24, 

the  product  24  being  obtained  from  8  just  as  3  is  obtained  from 
positive  unity. 

Similarly,      -  8  •  3  =  (-8)  +  (-  8)  +  (-  8)  =  -  24.  [§  16 

Again,  since  — 3  =  —  1  —  1  —  1,  i.e.,  since  —  3  is  obtained  by 
subtracting  positive  unity  three  times, 
therefore  8  •  (-  3)  =  -  8  -  8  -  8  =  -  24, 

and         _8.(-3)  =  -(-8)-(-8)-(-8)  =  8  +  8  +  8  =  24. 

Observe  that  two  of  the  above  products  are  positive  and 
two  are  negative.  How  do  the  signs  of  the  factors  compare 
when  the  product  is  positive?  when  it  is  negative?  How 
does  the  absolute  value  of  the  product  (cf.  §  15)  compare 
with  the  absolute  values  of  the  factors  ? 

*  Teachers  who  prefer  to  give  more  drill  on  addition  and  subtraction  at 
this  point  may  omit  §§  18  and  19  until  after  Chapter  III  has  been  read. 


17-19]  POSITIVE  AND  NEGAriVE  NUMBERS  25 

Applying  the  tlefinitioii  of  a  product  to  any  two  numbers 
whatever,  just  as  we  did  to  8  and  3,-8  and  3,  etc.,  above, 
we  see  that :  (1)  if  two  factors  have  like  signs,  their  product 
is  positive;  (2)  if  they  have  unlike  signs,  their  product  is 
negative;  and  (3)  the  absolute  value  of  the  product  equals 
the  product  of  the  absolute  values  of  the  factors. 

EXERCISE  X 

Find  the  following  products  and  explain  your  work : 
1.-5-2.  4.   9  .  -  3.  7.   3  .  -  4  .  -  5  .  2. 

2.-5.-2.  5.    -7|.-6.  8.   2'-'^x,i.e.,2'-^-x. 

3.-12.8.  6.-^.-5.  9.    _4«.3.-5.-7. 

10.  Show  that  ( -  2)\  i.e.,  _  2 .  -  2  •  -  2  •  -  2,  is  16.     What  is 

the  value  of  (-  2)'  ?  of  (-  3)^  •  (-  4)^  ? 

11.  What  is  the  sign  of  (-  7)^  ?  Why  ?  What  is  its  absolute 
value  ?     What  is  the  sign  of  (-  7)'^  -  2 . 5  ? 

12.  A  succession  of  multiplications  (as  in  Ex.  7,  for  instance) 
is  called  a  continued  product.  Can  the  sign  of  a  continued  product 
be  obtained  without  actually  performing  the  multiplication  ? 
How  ?     What  is  the  sign  if  there  are  5  negative  factors  ? 

13.  An  odd  power  of  a  negative  number  (i.e.,  a  power  whose 
exponent  is  odd)  has  what  sign  ?  An  even  power  ?  Is  a  power 
of  a.  positive  number  ever  negative  ?     Explain. 

If  a  =  —  4,  6  =  —  2,  c  =  3,  rf  =  —  1,  and  e  =  2,  find  the  value  of : 

14.  abode.  16.    ah^d'.  18.    If  —  c^—d^. 

15.  cHh\  17.    {a  +  hf.  19.    4.c'-cd^d\ 
Find  the  value  of  (a-\-h)  -  (x  —  y)  : 

20.  when  a  =2,  b  =  —3,  x  =8,  and  2/  =  —  5. 

21.  when  a=  —  4,  b  =  6,  x  =  3,  and  ?/  =  —  1. 

22.  when  a  =  ^,  b  =  —2a,x=  —  \,  and  y  =  \. 

19.  Division  of  algebraic  numbers.  Division  is  the  inverse 
of  multiplication  (cf.  §  8);  i.e.,  it  consists  in  finding  one  of 


26  HIGH   SCHOOL   ALGEBRA  [Ch.  11 

two  numbers  when  their  product  and  the  other  number  are 
given.  Hence  the  results  of  §  18  may  be  used  to  show  how 
to  divide  algebraic  numbers. 

Thus,  since  8-3  =  24,  8.(-3)  =  -24,  _8.3  =  -24,  and 
(_8).(-3)=24, 

therefore  24  --  3  =  8, 

_24-(-3)=8, 
_24-3=3_8, 
24  -  (-  3)  =  -  8. 

So,  too,  whatever  the  given  numbers.  Therefore  :  (1)  the 
absolute  value  of  the  quotient  of  two  algebraic  numbers  is 
the  quotient  of  their  absolute  values,  (2)  this  quotient  is 
positive  if  the  dividend  and  the  divisor  have  like  signs,  and 
(3)  it  is  negative  if  they  have  uiilike  signs. 

EXERCISE  XI 
Find  the  value  of  each  of  the  foljowing  indicated  quotients : 

1.  14-2.  4.    -31- (-If).  7.    15 -(-1). 

2.  14-- (-2).  5.    -24-9.  a    -365--(-9i). 

3.  _18^41  6.    (-6)2-(-2)3.         9.    _63a2^(-9). 

10.  Of  what  operation  is  division  the  inverse?  How,  then, 
may  the  correctness  of  a  quotient  be  tested  ?     Illustrate. 

11.  If  the  dividend  is  positive,  and  the  divisor  negative,  what 
is  the  sign  of  the  quotient  ?  Compare  the  signs  of  divisor  and 
quotient  when  the  dividend  is  positive ;  when  it  is  negative. 

Find  the  value  of  each  of  the  following  expressions : 

12.  24-28-(-7)+(-16)-(-4).(-3). 

13.  _8.(-6)--24-27--(-6)^3. 

14.  j28-(-7)-2.(-4-2)  +  24j-(-2)3. 

Verify  that =  —. -„ : 

•^  x-i-y     X  —  y      x^  —  y^ 

15.  when  a  =  Q,  b  =  2,  a?  =  10,  and  y  =  6. 

16.  when  a=  —  8,  b  =  12,  x=  —  9,  and  y  =  7. 


CHAPTER   III 

ADDITION  AND  SUBTRACTION  OF  ALGEBRAIC  EXPRESSIONS 
—  PARENTHESES 

I.   ADDITION 

20.  Algebraic  expressions,  monomials,  etc.  Any  combina- 
tion of  letters,  or  of  letters .  and  numerals,  representing  a 
number  is  called  an  algebraic  expression. 

The  terms  of  an  algebraic  expression  are  the  parts  into 
which  it  is  separated  by  the  signs  +  and  —  (or,  rather,  these 
parts  together  with  the  signs  preceding  them).  Thus,  3  a, 
+  m^,  and  —  5cx  are  the  terms  of  the  expression  3  a  +  w^  —  5  ex. 

An  algebraic  expression  which  is  not  separated  into  parts 
by  +  or  —  signs  is  said  to  consist  of  a  single  term,  and  is 
called  a  monomial.  Thus,  3  a,  5mx^,  and  —llcV  are 
monomials. 

An  expression  consisting  of  two  or  more  terms  is  called 
a  polynomial.     Thus,  S  a -{-7 m^—  5 ex  is  a  polynomial. 

A  polynomial  consisting  of  two  terms  is  usually  called  a 
binomial,  and  one  of  three  terms,  a  trinomial.  Thus,  2  s—5  xy 
is  a  binomial,  while  4  a;  —  a  +  7  ^^^  is  a  trinomial. 

21.  Coefficients.  Any  one  of  the  factors  of  a  term,  or  the 
product  of  two  or  more  of  them,  is  called  the  coefficient 
(co-factor)  of  the  product  of  the  remaining  factors.  Thus, 
in  the  term  5  axy^.,  the  coefficient  of  axy^  is  5,  the  coefficient 
of  x'lp'  is  5  a,  the  coefficient  of  5  xy^  is  a, 

A  coefficient  consisting  of  numerals  only  is  called  a  numer- 
ical coefficient,  while  one  that  contains  one  or  more  letters 
is  called  a  literal  coefficient.     Thus,  in  the  term  —  3  a^m^  the 

HIGH  SCH.  ALG.  — 3  27 


28  HIGH  SCHOOL  ALGEBRA  [Ch.  Ill 

numerical  coefficient  of  ax^m  is  —  3  ;  but  —  3  a  and  —  '6  am 
are  literal  coefficients  of  xhii  and  x?^  respectively. 

Eemakk.  The  word  "  coefficient "  is  usually  understood  to 
mean  ''  numerical  coefficient "  and  to  include  the  sign  preceding 
the  term.     Observe  also  that  ax  means  the  same  as  lax  (cf.  §  9, 

Note). 

22.  Positive  and  negative  terms.  Like  and  unlike  terms. 
A  term  preceded  by  the  sign  +  is  called  a  positive  term,  and 
one  preceded  by  the  sign  —  is  called  a  negative  term.  If 
the  first  term  of  an  algebraic  expression  is  positive,  its  sign 
is  usually  omitted,  but  the  sigli  of  a  negative  term  is  never 
omitted. 

Terms  which  either  do  not  differ  at  all,  or  which  differ 
only  in  their  coefficients,  are  called  like  terms,  also  similar 
terms ;  terms  which  differ  in  other  respects  are  called  unlike 
terms,  also  dissimilar  terms.  Thus,  3  x^y^  —  5  a^y,  and  |  x^y 
are  called  similar  terms. 

EXERCISE  XII 

1.  Name  the  coefficient  of  d^x  in  each  of  the  following  terms : 

3  a^x,    —  5  d^x,   dx.   4  dhx.  —  |  a^x.   —^ ,   —  9  a^x. 

2.  In  Ex.  1  which  coefficients  are  literal  and  which  numerical? 
Which  terms  are  positive  and  which  negative  ? 

3.  Do  the  positive  terms  in  Ex.  1  necessarily  represent  positive 
numbers  for  all  values  that  may  be  assigned  to  the  letters  in- 
volved ?     Try  a  =  3  and  a;  =  -  2. 

4.  What  is  the  coefficient  of  x  —  ym  each  of  the  follow- 
ing expressions :  13  (a?  —  y),  —  a  (x  —  y),  ^  m  (x  —  y),  and 
(4  —  a^)  (x  —  y)  ?  Which  of  these  coefficients  are  numerical  ? 
Which  literal  ? 

5.  Consult  a  good  dictionary  for  the  derivation  of  the  words 
"monomial,"  "binomial,"  "trinomial,"  and  " polynomial."  Write 
three  monomials,  three  binomials,  three  trinomials,  and  three 
polynomials. 


21-23]         ADDITION   OF  ALGEBRAIC  EXPRESSIONS  29 

6.  Distinguish  between  the  meanings  of  5  in  the  expressions 
5  X  and  oc^.     What  name  is  given  to  the  5  in  each  case  ? 

7.  What  are  like  terms  ?  By  what  other  name  are  they  known  ? 
In  what  respects  may  they  differ  and  still  be  like  terms  ? 

8.  Are  3  x^y,  —  2  x-y,  and  |  x^y  similar  ?  Are  4  ax^  and  —  6  bx^ 
similar  ?  Are  these  last  two  terms  similar  if  4  a  and  —6b  are 
regarded  as  their  coefficients  ? 

9.  Write  three  sets  of  like  terms,  some  terms  being  positive 
and  some  negative,  and  each  set  containing  at  least  four  terms. 

23.  Addition  of  monomials.  Since  5  times  any  given  num- 
ber plus  2  times  that  number  is  7  (^^e.,  5  +  2)  times  the 
given  number,  therefore  5a  +  2a=(5-f2)a=7a,  whatever 
the  number  represented  by  a.  Similarly,  8  mx^i/  +  8  mx^i/  — 
(3  +  S)mx^i/  =  11  mx^i/.     Hence, 

To  add  two  or  more  similar  monomials,  add  their  coeffieie7it>i 
and  to  this  result  annex  the  common  literal  factors,  each  with 
its  proper  exponent. 

It  is  usually  more  convenient  to  write  the  terms  to  be 
added  under  one  another,  as  in  arithmetic,  thus : 
3  XT/  153  ahnx  18  aks 

8  xi/  74  a!^mx  —  7  aks 

Wxy  '       ?  ?         (cf.  §  16) 

If  the  monomials  to  be  added  are  dissimilar,  they  cannot 
be  united  into  a  single  term,  but  their  sum  may  be  indicated 
in  the  usual  way  ;  thus,  the  sum  of  5  a  and  2  a:^  is  5  a  H-  2  a;^. 
Similarly,  the  sum  of  3m  and  —  6a  is  3m  +  (—  6  a),  which 
equals  3m  — 6  a  (cf.  §  16). 

EXERCISE  XIII 

Add  the  following  sets  of  similar  termsf  and  explain  your  work: 
1.  2.  3.  4. 

6  n  18  a^  —9  mx  31  abvr 

3  n  - 10  a^  5mx  -  22  abx" 

—  2n  —  3  a^  — 6  mx  —   6  aba^ 


30  HIGH  SCHOOL  ALGEBRA  [Ch.  Ill 

5.  State  a  convenient  rule  for  adding  any  number  of  like 
terms.  Does  your  rule  apply  to  cases  in  which  some  of  the  terms 
are  negative  ? 

Find  the  algebraic  sum  of: 

6.  4  a?^,  —  2  x^y  —  5  x^.  10.   12  a^n,  a^n,  —  4  a%  —  9  ahi, 

7.  11  ax,  ax,  —  9  ax.  11.   S  xz,  —  8  xz,  —  xz,  2  ics;. 

8.-4  cs^,  —  cs^,  8  cs^.  12.    —  a6^,  —  7  a6^,  a5%  —  5  ah*. 

9.    —  3  aa;"2/j  ''^  cl^Vj  ctx^y.        13.    —  ic^/?  ~  ^  ^I/)  12  a;?/,  —  3  a;y. 

Simplify  the  following  expressions;  i e.,  unite  like  terms,  and 
indicate  the  results  where  the  terms  are  unlike : 

14.  3  bxy^  +  (-  4  bxy^)  +  (-  12  bxij^  +  5  bxy^  +  bxy^  +  (-  bxy^. 

15.  —  4  mp^  +  13  a'^x  +  7  mp^  -f  3  mp^  -f  (  —  5  ax^  —  2  acc^  +  mp^ 

16.  25  c^s^^  - 10  6^^  -  bH  -  c^s^  -\-3bH-{-  bH  +  cV  -  8  (fs\ 

17.  7.5a;  +  fx  —  a;  —  i£C  +  iic  +  i^x  —  3.45  x  +  li  a;. 

18.  3d*-5  c/  +  2ic^^-11.5d^-7ic^'^-5d^  +  d*-c^. 

19.  -  6 (a-  62)  4-3  (a-  62)  _  («_  ^,2)  _  5  (ct_  52^  j^^a-b"), 

20.  23  a^  +  5  6"  -  8  a^^**  - 13  6«  +  24  a^^"  _  19  a^  +  a^  -  6"  -  a^ft". 

21.  How  many  a^'s  in  5  a;  +  3  a;  ?     in  10  a?  —  2  a;  ?     in  8  a;  —  aa;  ? 
in4aj  — aj?     ina;  — Sa^  +  lla;?     in  ma? -f  na;  —  3  a;  ? 

22.  How  many  s^^'s  in  8  s^^  +  2  sH  ?     in  3  sH  +  as't  ?     in  3^  s^^ 
+  2  ms^^  -  s2^  ?     in  3  asH  +  (-  2  fts^^)  ? 

23.  Add  15  a;2,  -  2  aa;2,  -  7  a^^^  4  fta^^^  and  -  2  Za;^. 
[In  Exs.  23-25  let  «,  6,  and  ^ belong  to  the  coefficients.] 

24.  2  aa;?/)  ~  8  ^2/>  ^^^  ^  ^^2/-  25.    5  xz^,  —  axz^,  and  2  bxz^. 

24.   Addition  of  polynomials.     Any  two  polynomials,  e.g.^ 

S  a^  —  7  xi/  -}-  12  7/^    ai^d    5  a^  4-  6  2:3/  —  3  «/2,    may   be    added 

thus: 

3a2_7^y  +  12^2 

5  a^  _|_  (5  2;^  _    3  ^2 

8a2_     a;y+     9?/2 


28-26]         ADbtriON   OF  ALGEBRAIC  EXPRESSIONS  31 

This  procedure  may  be  stated  tliiis :  To  add  two  or  more 
polynomials^  write  them  under  07ie  another  so  that  similar  terms 
shall  stand  in  the  same  column,  and  then  add  each  columii  sep- 
arately as  in  §  28. 

EXERCISE   XIV 

Add  the  following  sets  of  polynomials  : 


1. 

2. 

3. 

4. 

4a-26 

6m 

4-3^2 

ax  — 5 

y 

8|)  +  2.s-3^2 

2a4-56 

4m 

-7  7l' 

2ax- 

1 

Wp-bs-^lt^ 

5. 

6. 

7. 

6  m  —  4  ^i 

+  7p 

Sa 

-40^  +  5?/ 

-  a^a;  -  8  6  -h  2  / 

-  2m+     n 

-5p 

6a 

-i-5x'-    y 

—  2  a?x           —^'f 

-Sm-{-2n 

-4.P 

-4.a 

^2^ 

8  a-x  +  2  6  -  9  .v*" 

8.  12  ace  —  5  a?^  —  9  2/,  —  3  ao?  —  6  a^^  +  2  ?/,  and  ax  +  x'^'  —  y. 

9.  3  m  —  7  71  +  2p,  m  H-  4  ??  —  6p,  and  n  —  2m  +p. 

10.  a^  —  2  .T  + 1,  a;  —  3  +  8  a^,  and  4  a;  —  3. 

11.  2a-76  +  3(a^-l),  4.b- a-6{x' -I),  and  6+  (a;--l). 

12.  3(a  +  &)  -  2  c'  -  5,  7  -  6{a  +  &)  +  4  c',  and  c'  -  (a  +  h). 

13.  2(?7i  -  n)  +  4(|>  + 1),  3  a  —  5(m  —  w),  and  —  7{p  +  1)  —  9  a. 

14.  2f  A:-6.5?  +  3im,  5fc-6im,  and4m  +  2Z. 

15.  s''-^t-^v,2t-^s\^.\i^i^v-^t+2s\ 

Supply  the  missing  coefficients  in  the  following  equations : 

16.  12a;  — 4a?/  — 5aj-h7a2/=  ^x  +  '^ay. 

17.  aoi?  —  2  xy -\- dxy  —  CO?  =  ?  x^ -\-  ?  xy. 

18.  6rs-s^  +  5cs3+(2-3c)ns=?rs+?s3. 

19.  (a2-c)p  +  (3a4-5c-5)i?=?/>. 

20.  Add  3  a?2  +  4  a;/  —  2/"*  —  7  a;?/  +  2  a:^?/,  10  a^y  -f-  (5  c  -  10)  xy-, 
(c  + 1)  a:^  —  3  2/%  and  xy -\- ^  xy"^ -{■  {2  d  —  l)x^y. 

25.  Checking  results.  If  a  result  (iu  addition,  for  example) 
is  correct,  then  it  must,  of  course,  remain  correct  when  we 
assign  any  arbitrary  values  to  its  letters.     This  is  the  basis 


32  BIGH   SCHOOL   ALGEBRA  [^'h.  Ill 

of  a  very  useful  test  of  the  correctness  of  algebraic  work 
(usually  called  a  ''  check  "). 

Thus,  find  the  sum  of  10  a^  -  3  7/  and  -  2  a;^  _^  9  2/. 

SOLUTION  CHECK 

.       10.^-3.,         =    10-    6=.   4  1      j^^^^^^^ 

-^^  +  ^^         =-2  +  18=26  L,d     ,=  2 
80^  +  62/  =     8  +  12  =  20  J 

And  since  the  sum  of  4  and  16,  the  values  of  the  summands,  is 
20,  which  is  also  the  value  of  the  sum,  therefore  the  work  is  prob- 
ably correct. 

EXERCISE  XV 

In  Exs.  4-11,  p.  31,  check  your  results  as  above  by  putting  any 
convenient  arbitrary  values  for  the  letters. 

II.    SUBTRACTION 

26.  Subtraction  of  monomials.  Since  subtraction  is  the 
inverse  of  addition,  therefore  (cf.  §  23)  : 

To  suhtraet  one  of  tivo  similar  monomials  from  the  other,  sub- 
tract the  coefficient  of  the  subtrahend  from  that  of  the  minuend^ 
and  to  this  remainder  annex  the  common  literal  factors. 

The  work  may  be  arranged  thus : 

73  a2  27n^k  mx^f  Uax 

24  a2  -Sn^k  19  2;^  -  9  ax 

49rt2  ~J5M  ~1  ~~f~  (cf.  §  IT) 

If  the  given  monomials  are  dissimilar,  the  subtraction  can, 
of  course,  only  be  indicated. 

A  good  practical  rule  for  subtraction  is :  To  subtract  one 
of  tivo  similar  monomials  from  the  other^  reverse  the  sign  of  the 
subtrahend  and  proceed  as  in  addition.  In  order  to  avoid 
confusion  in  reviewing  one's  work,  it  is  best,  however,  not 
actually  to  reverse  the  written  sign  but  only  to  conceive  it 
to  be  reversed. 


25-27]     SUBTRACTION  OF  ALGEBRAIC  EXPRESSIONS  38 

EXERCISE  XVI 

Perform  the  following  indicated  subtractions : 
1.    18-5;   -18-5;    _18-(-5);    18-(-5);    9-(-9). 
2.  3.  4.  5.  6. 

7a  16ba^  - 18  m^  -ISr^a^  26  ^V 

4  a  -Sbx^  Im^  -    Tr^cc^  _  9  ^y 

7.  8.  9.  10. 

3  ex"  -  6  7nY  6. 8  k'a^y'-  -  21  a^m* 

11.  Show  that  "  changing  the  sign  of  the  subtrahend  and  pro- 
ceeding as  in  addition"  will  give  the  remainder  in  each  of  the 
above  exercises. 

12.  From  7  aoc^y  take  3  ax-y ;  from  5  njf  take  —  8  7q)^ ;  from 
4  (a  -  2  b^)  take  -  11  (a  -  2  ¥) ;  from  the  sum  of  13  2/"V  and  -  5  y-^'^ 
take  4  i/^a;^. 

13.  Indicate  the  subtraction  of  b^  from  3  a-;  of  4(0^  +  ?/^)  from 
—  6  (c  +  2/)  ;  of  2  a;?/  from  the  sum  of  x^  and  2/" ;  of  —  a^ft"  from  the 
sum  of  3  a*  and  —  6^**. 

14.  How  many  x^y's  in  8  aic-^/  —  2oify?  in  mxry-{-nx-y—2  cx^y  ? 
in  7  cx'y-(- 3  x'y)? 

15.  Supply  the  missing  coefficient  in  :    125 7nz  —  97 mz=  ?  Z', 

c2a2-(-9a2)=?a2;    5  ax^  +  ?  ao;^  =  2  aa;=^ ;    4:C8-?s  =  2bs. 

27.  Subtraction  of  polynomials.  One  polynomial  may  be 
subtracted  from  another  by  writing  the  subtrahend  under 
the  minuend,  similar  terms  under  one  another,  and  subtract- 
ing term  by  term,  thus : 

55_3^2+    Q^l  32:4-5 

85  + 5:^2  _    9^^  _7a;4  4-4-2:?: 

2  6  -  8  ^2  _^  15  ab  10ai^'^)-\-2x 


34  HIGH  SCHOOL  ALGEBRA  [Ch.  Ill 

EXERCISE  XVII 

In  the  following  pairs  of  expressions  subtract  the  second  from 
the  first,  and  check  your  results  as  in  §  25 : 

1.  8a-562,   2a  +  h\  3.    ^x'  +  x,   Zx-2x', 

2.  3m2-7,   m^-lO.  4.    s'  +  ^t,   2t-5s', 

5.  a^-2ab  +  b',    -3  a' +  12 ab-12b\ 

6.  x^-\-5oi^y  +  7  xif  —  2y^,   3  ic/  —  ar'  —  2  /  —  5  a?y. 

7.  Check  your  answer  to  Ex.  6  by  letting  x  =  2  and  y  =  l. 

8.  From  the  definition  of  subtraction  show  that  the  minuend 
equals  the  sum  of  the  subtrahend  and  remainder.  What  means 
of  checking  the  result  does  this  suggest  (cf .  §  17)  ? 

9.  In  each  of  Exs.  1-6,  p.  31,  subtract  the  second  expression 
from  the  first. 

10.  From  c^  -\-d  subtract  c^  —  d  —  4  A;.    Check  result  in  two  ways. 

11.  From  5  x^  +  4  a'b  take  8  a^6  —  2  x^  +  5  abx,  and  check  result. 

12.  Subtract  15  y^  + 10  aV  +  4  mV  from  34  aV  - 10  mV,  and 
check  result. 

13.  Subtract  15  —  3  a?  +  10  aj^  from  12  a^  +  5  j  also  from  —  2x; 
also  from  0. 

14.  Subtract   5^  a^-2i  +  a;-4J  a;^  from  7  a;^~2i- a;  +  a^-4; 
also  from  0. 

15.  Subtract  the  sum  of  5  a  —  31  6^  +  2  a^  and  26  6^  —  4  a;  from 
ix^-2a'  +  7b\ 

16.  From  what  must  a^b""  +  3  ca;  +  cZ*"  be  subtracted  if  the  result 
is  to  be  4  a^b''  —  d'-^2cx? 

In  Exs.  17-20  let  a,  b,  and  m  belong  to  the  coefficients : 

17.  From  2ax  —  Sby-\-  mxy  take  x  —  2y-\-^^xy. 

18.  From  (m - 2  6) ?/  +  3\z  take  2z+(a-  b)y\ 

19.  From  Qr  -l)xy -\-  bY  -  3  mx^  take  2  V'xy  -  5  mx^  +  af. 

20.  From    (a^  -  3  a6  +  m^)^  -f  (4  a^  -  5  «?>  +  2  6*'  +  7  m^)  ?/  + 
2  amz^  subtract  {cir  -  5  o6  +  &^  -  m')^  -  3|-  a^62^  +  (a^ -  2  a6  +  m")  y. 


27-28]  PARENTHESES  35 

III.    PARENTHESES* 

28.  Parentheses  removed  and  inserted.  Such  an  expres- 
sion as  2  a;  —  (?/  —  3  0)  means  that  i/  —  3  2  is  to  be  subtracted 
from  2  X  ;  hence  (§  27) 

2x-(iy-Zz)  =  2x-y  +  ^z, 
Similarly,    a  —  (^—h  +  c—  d  —  e)  =  a-\-h  —  c  +  d-{-  e;  etc. 

These  equations  (read  from  left  to  right)  show  that  a 
parenthesis  inclosing  any  number  of  terms,  and  preceded  by 
the  minus  sign,  may  be  removed  provided  that  the  sign  of  each 
term  within  the  parenthesis  is  reversed. 

Again,  reading  the  above  equations  from  right  to  left 
shows  that  any  number  of  the  terms  of  an  expression  may 
be  inclosed  within  a  parenthesis  preceded  by  the  minus  sign, 
provided  that  the  sign  of  each  term  so  inclosed  is  reversed. 

Remark.  A  parenthesis  preceded  by  the  plus  sign  may,  of 
course,  be  removed  or  inserted  without  changing  the  signs  of  the 
terms  inclosed.     (Why  ?) 

EXERCISE  XVIII 

By  means  of  §  27  show  that : 

1.  5a  —  (3a  +  ^)=5a  —  3a  —  &  =  2a  —  &. 
[What  is  the  sign  of  3  a  in  (3  a  +  6)  ?] 

2.  ^x  —  2y—{—^x-\-y)=^x  —  2y-{-^x  —  y=zlx  —  Sy. 

3.  m^  —  3  np  4-p^  =  m^  —  (3  np  —p^). 

4.  a  —  2b-^c  —  4.x  =  c  —  4:X-{—a-\-2b). 

5.  Using  §  11,  show  that  8  -  (10  -  7)  =  8  -  3  =  5 ;  and  then 
show  that  the  same  result  may  be  obtained  by  using  §  27,  i.e., 
show  that  8-(10-7)=8-10  +  7  =  -24-7  =  5. 

Simplify  each  of  the  following  expressions  by  two  methods, 
,  as  in  Ex.  5,  and  compare  results : 

*  "  ^Parenthesis  "  here  means  any  sign  of  aggregation  whatever  (cf.  §  11). 


36  HIGH  SCHOOL   ALGEBRA  [Ch.  Ill 


6.    ll_(3  +  6).  9.   27  +  (-5-3)-13-7. 


7.   ii_(_3  4.6).  10.   27 -(-5 -3) +  13 -7. 

a    _(8-5)  +  10.  11.    -(6-4  +  9)+3-(-2  +  7). 

12.  In  Ex.  9  what  is  the  quality  sign  of  13  ?  What  does  the 
minus  sign  preceding  13  indicate  ? 

In  Exs.  13-19  remove  parentheses  and  unite  similar  terms : 

13.  7x  —  Sac-\-(x  —  2 ac).     [Compare  §  28,  Remark.] 

14.  3a  — 46 +  (6  — 2a).  17.   x  —  y  +  {x-\-y)  —  {^x  —  y). 

15.  2f-{-x'  +  f-xy).         18.    a-2/2-(a-3)-(-3/-l). 

16.  5a2  +  3?>-(-2a).  19.    -(2m-5)-(-6+a^-3m). 

In  each  of  the  following  examples  inclose  the  last  two  terms 
in  a  parenthesis  preceded  by  —  : 

20.  2s-3^  +  w.  23.    aa^  — 4  5x  — 3  +  2^. 

21.  6  4-5  X-  — 3?/.  24.   2 /i  — 3  A;  — 7  a?  — 5. 

22.  a^-f2  6  +  c*.  25.    3  m*  —  2  m"ic  —  5  ma^"  +  a;*. 

29.  Parentheses  within  parentheses.  It  often  happens 
that  one  parenthesis  incloses  one  or  more  others.  In  such 
cases  the  expression  within  an  inner  parenthesis  forms  a 
single  term  of  the  next  outer  parenthesis  [cf.  §  11  (ii)]. 

These    parentheses,    too,    may   be    removed   as   in   §    28, 
thus : 
3  a^  -  J  9  m  -  [  -  a^  -  (4  s3  _  5  ^^^)  ^  ,,3-|  j 

=  3  a^  —  9  m  -f-  [  —  a^  —  (4  s^  —  5  m)  +  s^]      [Removing  brace 
=  3  a^  —  9  m  —  a^  —  (4  s^  —  5  m)  -+-  s^  [Removing  bracket 

=  3a^  —  9m—  a^  —  4s*  +  5m  +  s^      [Removing  parenthesis 
=  2  a^  —  4  m  —  3  s'.  [Collecting  terms 

Let  the  pupil  simplify  the  above  expression  by  first 
removing  the  innermost  parenthesis,  then  the  next  inner- 
most, and  so  on,  and  compare  his  work  with  what  is  here 


28-29]  PAIiENTHESES  37 

EXERCISE  XIX 

Simplify  the  following  expressions;  that  is,  remove  the  paren- 
theses and  combine  like  terms  : 

1.  s  -  [t^ -\- (u'^  ~  s)'].  5.  (m  —  4p)  -  (a  —p  +  m). 

2.  s-[e-(ir-s)].  6.  Sx'-\2a-(-x'-{-a)l, 

3.  6a-[6-(-2a-|-36)].  7.  _  J2a  -  (-ic^.^,  ^^  j^ 

4.  a^^l-f-(2f-3x')l  8.  -J_(-a^  +  a)J. 
9.  mx^— [8?/— (6a;  — W.1')  —  2aj. 

10.    _  (60  -  25) -[92 -(18 +  27) I . 


11.   3p  +  4g  +  [7i)-2g-(5i)-3-5g)]. 


12.  a  — y— \a  —  (—y  — a  — 2)1. 

13.  a;-S3a.--[-(-3a;  +  22/)+o?/]-32/: 


14.    8a;2^2a;?/-[3x2-?/2-(2a;.?/-.T-  +  /)]-2/2. 


15.  2  a'  -  5  [3  6"  -  (a'  -  2  c  -  5  a')]  _  [7  6"  +  5  c  -  6«  -  2  a']  \ . 

16.  8a-25-{-(3c-f?)-[4c-d-(-8aH-2&)]-2d;. 

17.  4.-[5y-l3-(2x-2)-4.xl'}-  Ja;  +  52/-^+3J. 

18.  In  Ex.  2,  how  many  minus  signs  affect  u^?  How  often, 
then,  will  its  sign  be  reversed  by  removing  the  parentheses  ? 
What  will  be  its  sign  finally  ?  Answer  the  above  questions  for 
-  «2  in  Ex.  7. 

19.  By  considering  the  number  of  minus  signs  affecting  the 
respective  terms,  remove  together  all  parentheses  in  Ex.  9.  Also 
in  Exs.  11,  12,  and  14. 

20.  In  the  expression  3  m  —  4  a  -f- 10  a;^  —  5  ?/  +  3  a^^  —  8  aa;,  in- 
close the  4th  and  5th  terms  in  a  parenthesis  preceded  by  the 
minus  sign ;  then  inclose  this  parenthesis,  together  with  the  two 
preceding  terms,  in  a  bracket  preceded  by  the  minus  sign.* 

21.  Make  the  changes  asked  for  in  Ex.  20,  in  the  expressions 
3m  +  4a-10a;2_5^_^3^^2_g^^^  3m-4a -2a?2-h52/-3a6-, 
and  —  5  a;''  +  3  ?/'"  -  4  a  -  14  6c  +  8  A;2. 

*  The  value  of  the  expression  is,  of  course,  to  be  left  unchanged. 


CHAPTER   IV 

MULTIPLICATION   AND   DIVISION   OF   ALGEBRAIC   EXPRES- 
SIONS 

L    MULTIPLICATION 

30.  Law   of  exponents   in   multiplication.      What   is   the 

meaning  of  5^  (cf .  §  9)  ?    of  o^  ?    of  a;"  ? 

How  many  times  is  s  used  as  a  factor  in  the  product 
8^-8^?     Is  8^  '  8^  equal  to  8^  ?     Explain  why. 

How  may  the  exponent  of  the  product  ^  -  8"^  he  obtained 
from  the  exponents  of  the  factors  s^  and  8^  ?  Would  your 
answer  remain  true  if  we  were  to  put  other  exponents  in 
place  of  3  and  2  ? 

Is  cc^x^a^  equal  to  x^'^,  i.e.,  to  2^^+2+5?  W^hy  ?  Is  x"x^  equal 
toaj«+*?     Why? 

The  results  of  these  considerations  may  be  expressed  in 
symbols  thus : 

wherein  a  may  stand  for  any  number  whatever,  but  m,  w, 
and  p  are  positive  integers. 

Translated  into  common  language,  this  law  of  exponents  is  : 
The  product  of  two  or  more  power8  of  any  number  is  that 
power  of  the  given  number  who8e  exponent  i8  the  8um  of  the 
exponents  of  the  factors. 

31.  Product  of  two  or  more  monomials.  The  product  of 
any  two  or  more  monomials  may  be  obtained  by  a  simple 
extension  of  §  30. 

J7.^.,  in  the  product  of  2  ax^  and  3  ab'^x,  how  many  numeri- 
cal factors  ?     What  is  their  prodilct  ?     How  many  a\s  in  the 

38 


30-31]     MULTIPLICATION  OF  ALGEBRAIC  EXPRESSIONS     39 

entire  product?  How  many  5's  ?  How  many  a;'s  ?  ^\^rite 
down  this  product,  using  the  exponent  notation.  What  is 
its  sign  ?     Why  ? 

What  is  the  product  of  3  aV^  —  2  abaP'^  and  5  «5  ?  Is  it 
—  30  a'^h'^x^  ?  Explain  in  detail,  mentioning  the  sign,  the 
coefficient,  the  letters,  and  their  exponents. 

These  considerations  lead  to  the  following  rule  for  obtain- 
ing the  product  of  two  or  more  monomials:  To  the  product 
of  the  numerical  coefficients  of  the  several  monomials,  annex  the 
different  letters  which  these  monomials  contain,  giving  to  each 
letter  an  exponent  equal  to  the  sum  of  the  exponents  of  that 
letter  in  the  several  monomials. 

EXERCISE  XX 

Find  the  following  indicated  products,  and  explain  your  re- 
sults, especially  the  signs  and  exponents: 

1.  2.  3.  4.  5. 

3a2  3a2  6-3^         -Smh         -7aV 

5a^  -5a^         -2.3^         -3ms^         -2  2^ 

6.  7.  8.  9.  10. 

-5-23  mV  2  a;"  -Sa^^"         _4a;2y 

8.2^  -ba^x        -QxP  -Sab^  -9x^y^ 

11.  7  s2 .  _  3  as  .  -  2  a\  14.    -  2  a"6  •  6f  ab^'- 1 2|  a'^bK 

12.  3  mx^  •  2  m^  •  —  7  am.  15.   7.5  mv'^  -  —  4|  am^  •  —3  a"^. 

13.  _  2i  s? .  -  s^ .  3f  a^^.  16.    -  12  A;^/^ .  _  aA:Z"  •  2^  a"A:^ 

17.  2  (a -6)2.  -  5  (a  -b)  '  3 x'y  '  -2^ x  (a-b)  .2.5(a-6)Y. 

18.  Define  and  illustrate  the  meaning  of  exponent,  power, 
base  (of.  §  9).  May  the  base  be  negative  ?  May  it  be  a  frac- 
tion ?     May  the  exponent  be  fractional  or  negative  ? 

19.  If  7z  represents  a  negative  number,  is  n^  positive  or  nega- 
tive (cf.  Ex.  13,  p.  25)?  How  does  3^  compare  with  (-3)^? 
2^  with  (-2)^? 

20.  What  is  the  meaning  of  2/"~^  ?  In  this  expression  may  n 
be  less  than  2  ?     What  is  the  product  of  4  a^  and  —Ta""-^? 


40  HIGH  SCHOOL  ALGEBRA  [Ch.  IV 

21.  Determine,  by  inspection,  the  sign  of  the  result  in  each  of 
the  following  products  when  a  =  —  2,  6  =  3  :  (a  —  6)^ ;  (a  —  6)^ ; 
(a  +  hy ;  {att^Y ;  (a^b^y ;  (a^  —  by.     State  your  reason  in  each  case. 

32.    Product  of  a  monomial  and  a  polynomial.     From  the 

definition  of  a  product  [cf.  §  7  (ii),  §  18], 

5  .  (2  +  9)  =  5  .  2  +  5  .  9, 
since  2  +  9  is  obtained  by  taking   positive   unity  2  times, 
then  9  times,  and  adding  the  two  results. 

Similarly,  a{m  -\-  x—  y)  =  q^i  -\-  ax  —  ay^ 

whatever  the  numbers  represented  by  a,  m,  x,  and  y. 

Hence  we  may  say  :  To  find  the  product  of  a  polynomial  and 
a  monomial  multiply  each  term  of  the  polynomial  by  the  mono- 
mial and  add  the  partial  products. 

The  actual  work  may  conveniently  be  arranged  thus  : 

Check 
3  a%- 4x2 +  11 2/2  =      10 

-2xy =    -  2 

-  6  aH^y  +  8  x^y  -  22  xy^  =  -  20 

EXERCISE  XXI 


when  a  =  1,  X  =  1, 
and  y  =  1 


1.   How   is  2  -\-a  —  x  obtained  from  + 1  ?     How,  then,  may 
2/(2  +  a  —  ic)  be  obtained  from  y  ? 


2. 

3. 

4. 

Multiply 

3x-5y 

a-4a6  +  362 

2  m  —  Sn^  —  mn 

by 

2x 

-56 

—  4  mn 

Find  the  following  products,  and  check  results  (cf.  §  25) : 

5.  (2a^-4:y')'7xy.  9.    -Sx^y^(2  x"" -4:X^y). 

6.  (4:ax-5xy)(-2x^).  10.   5  a\a^-~4  a^ -2  a-\-7), 

7.  (~2s^-Sst)(-9st%  11.    -7ax'(Sax^-3a'x-5ax). 

8.  {-6u'-\-nv)(-4:v'x).  12.    -  8  a^^"  (3  -  4  a"*  +  12  6"). 

13.  -12xy{2}x^-5lx-4:). 

14.  2iabc(1.5a'-{-7.5ab'-6b'). 


31-33]     MULTIPLICATION  OF  ALGEBEAIC  EXPRESSIONS    41 

15.  {^x'z-^x^^-4.xz''-^xz  +  ll){-^xz). 

16.  [a^+(a  +  iy4-(a-l)ic  +  l].2aa:2_ 

17.  («/  -  2  xy  -.15  a;y  +  4  ary  -  7  a?y  (-  cc^^/^-s). 

18.  2  (%2-3  i;)  [(^2  -  3  i;)^  - 13  a;  (?^2_3  '^^+2  a;2(^2_  3  ^,)_l]. 

19.  3a[7ar^-4(2a;2  +  aaj)  +  «(2a-3a;  +  l)]. 

20.  Multiply  da—bh-\-c—x  —  y  by  — 1,  and  show  that  the 
result  agrees  with  §  28. 

21.  By  what  nust  x  —  ly  —  2az  be  multiplied,  to  obtain  the 
product  12  ah:  —  14  a^y  —  4  ah  ? 

22.  Find  a  monomial  and  a  polynomial  whose  product  is 
6  ax" -10  a'x  - 14  aV  4-  8  aV. 

23.  Are  the  values  of  m  and  n  in  Exs.  9,  12,  and  18,  limited  in 
any  way  ?     If  so,  how  ? 

33.  Product  of  two  polynomials.  Since  m  +  ^  is  obtained 
by  taking  positive  unity  m  times,  then  n  times,  and  adding 
the  two  results,  therefore  (of.  §  32) 

(a -\- h  ■{-  c)  '  (m -\- n)  =  (a -\- h  -\-  c)m  +  (a  +  5  +  c)n 
=  am  -\-hm+  cm -\-  an -{- hn -\-  en. 

Similarly  for  any  polynomials  whatever ;  i.e.^  the  'product  of 
two  polynomials  is  obtained  by  multiplying  each  term  of  the 
multiplicand  by  each  term  of  the  multiplier^  and  adding  the 
partial  products. 

If  any  two  or  more  terms  of  a  product  are  similar,  they 
should,  of  course,  be  united. 

Such  a  multiplication  and  its  check  may  be  arranged  thus : 

Check 

a2  +  2  a&  -  62  =  +  2, 

a+  h =+2. 

(a2+2a&-&2).a=  a^  +  2a%-    aW' 

(a'^  +  2a&-62).6=:  ggft  +  2  a&2  _  &8 

a3  +  3a-26  _f.    aW-  -  63  =  +  4, 


when  a  —  \ 
and     6  =  1 


Remark.  The  product  of  three  or  more  polynomials  may  be 
obtained  by  multiplying  the  product  of  the  first  two  by  the  third, 
this  product  by  the  fourth,  and  so  on  [cf.  §  10  (1)]. 


42  BIGB  SCHOOL  ALGEBRA  [Ch.  IV 

EXERCISE  XXII 
Perform  the  following  indicated  multiplications : 

1.    {x+2d)'(x-a).  9.    (x^-xy  +  y^)'(x-y). 

.2.    {Sa^-5x)'(2a''-{-3x).     10.    (a^ -xy -i-y^)  -  (x  +  y). 

3.  (7-2m3).(3-5m=^).         11.    (0^-2  ay -\-y^)  •  {a- y). 

4.  (a'A-ab-\-b')'{a-b).        12.    (5  s^  -  2  i^^^ .  (5  ^3  _^  2  ^^^^ 

3.    (2s-Sf)-(Ss-4.t).  13.  (7ey2-2c/«).(7ey2  +  2^). 

6.  {x-{-a)'(x-{-a),  14.  (-4/g- 6  r)  •  (3^)-^?^)^ 
I.e.,  (ic  +  ay.  15.  (m^  —  w^  +  p^)  •  (5  m  —  2  np). 

7.  (3m2-10)l  16.  (2a;-32/  +  0-4)2. 

8.  {Gx'-Sayy.  17.  (a^- 3  a6  + 62_2  c)^. 

18.  (3m2  — 2mn)  •  (5m +  3w^)  =  ?  Check  your  result  by 
letting  m  =  1  and  n  —  1.  If,  in  the  product,  the  exponent  of  m 
should  be  wrong,  would  this  check  reveal  the  error  ?  Explain. 
Would  the  error  be  revealed  if  m  were  taken  equal  to  2  ? 

Multiply  (and  check  your  work)  : 

19.  m*  —  2  m^  —  6  m^  +  m  —  1  by  3  m^  +  m  —  2. 

20.  2ii?  —  1xy  +  Sx^  —  4.x-\-2y-\-lhjxy  —  Zx  —  2y. 

21.  a2-62_c2_2a6-25c+2acby  a-26  +  c. 

22.  I.^x^-2xy-2.3y^-^x-2.^yhy3ly-l^x. 

23.  x""  +  y""  hy  X  —  y '^  hj  y? -\- y'^ ',  by  a?"  —  i/**. 

24.  ic""  —  3  a;'"~^y  +  3/"*  —  3  x?/'"-^  by  a?^  —  2  a;?/  +  /. 

25.  2x+(3n  — l)?/by  (n  4-l)a^  — (3n  +  l)y. 

26.  r2-(2ri-«2)  by4?2_4  (^2r^-r2)-l. 

34.  Degree   and  arrangement  of  integral  expressions.     In 

multiplications  with  polynomials,  and  elsewhere,  it  is  often 
advantageous  to  arrange  the  terms  of  a  polynomial  in  a  par- 
ticular order;  such  arrangements  will  now  be  explained. 
-^  A  term  is  said  to  be  integral  if  it  contains  no  letters  in  its 
denominator ;  it  is  integral  in  a  particular  one  of  its  letters  if 
that  letter  does  not  appear  in  its  denominator.    A  polynomial 


83-35]     MULTIPLICATION  OF  ALGEBRAIC  EXPRESSIONS    43 

is  integral,  or  integral  in  a  particular  letter,  if  each  of  its 
terms  is  so. 

U.g.,  3  ax^  +      ^^  —     ^f  ^  is  integral  in  6,  m,  x,  and  v  ; 
a  3 

it  is  fractional  in  a;  its  first  and  last  terms  are  altogether 

integral,  while  its  second  term  is  integral  only  in  5,  m,  and  ^. 

e^  By  the  degree  of  an  integral  term  in  any  letter  (or  letters) 

is  meant  the  number  of  times  this  term  contains  the  given 

letter  (or  letters)  as  a  factor.     Thus,  7  a^x^  is  of  the  second 

degree  in  x,  and  of  the  fifth  degree  in  a  and  x  together. 

^The  degree  of  an  integral  polynomial  is  the  same  as  the 

degree  of  its  highest  term.     Thus,  Sa^  —  5  a^jp^y  —  2  bx^y^  is 

of  degree  4  in  a;,  3  in  ?/,  and  5  in  a;  and  7/  together. 

A  polynomial  is  said  to  be  arranged  according  to  ascending 
powers  of  some  one  of  its  letters  if  the  exponents  of  that 
letter,  in  going  from  term  to  term  toward  the  right,  increase  ; 
and  that  letter  is  then  called  the  letter  of  arrangement.  If 
the  exponents  of  the  letter  of  arrangement  decrease  from 
term  to  term  toward  the  right,  the  expression  is  said  to  be 
arranged  according  to  descending  powers  of  that  letter. 

Thus,  2x^  —  5  ax^y  —  7  Pxy^  4-  3  m^y^  is  arranged  according 
to  descending  powers  of  x,  and  ascending  powers  of  y, 

35.  Multiplication  in  which  the  polynomials  are  arranged. 

If  each  of  two  polynomials  is  arranged  according  to  powers 
of  some  letter  which  is  contained  in  each,  then  their  product 
will  arrange  itself  according  to  powers  of  that  letter,  and  the 
actual  multiplication  will  take  on  an  orderly  appearance. 

E.  g. ,  Ito  6nd  the1>roduct«l  7  ic  -  2  ic^  +  5  +  x^  by  3  ic  +  4  a;2  -  2,  arrange  the 
tft)rk<to«B: 

Check 

a-3-  2x2+    7x  +    5  =11/ 

4  x2  +  3  x   -    2  =5, 


4  a;5  -  8  x*  +  28  x3  +  20  x2 

3x*  -    6x3  +  21x2  +  15  X 

-    2  x3  +    4  x2  -  14  X  -  10 

4x6  _5a4  + 20x3  + 45x2  + X- 10        =55, 

HIGH  SCH.  ALG.  — 4 


when  X  =  1 


44  niGIl  SCHOOL  ATMEBJIA  [Ch.  TV 

EXERCISE  XXIII 

1.  Is  1^  a^oc^  integral  or  fractional  ?     In  what  letters  is  ' :;^ 

integral  ?  '   In  what  letters  is  it  fractional  ? 

2.  May  an  integral  term  have  a  fractional  coefficient  ?  Illus- 
trate.    Write  a  term  integral  in  m  and  n  and  fractional  in  d. 

3.  When  is  a  polynomial  fractional  in  a  particular  letter^? 
Write  a  binomial  fractional  in  a  and  b ;  a  trinomial  integral  in  a 
and  b,  but  fractional  in  c. 

4.  In  Ex.  11  below, 

(1)  Of  what  degree  is  each  term  of  the  multiplicand  ?  each 
term  of  the  multiplier  ? 

(2)  Of  what  degree  in  a?  is  the  first  term  of  the  multiplicand  ? 
Of  what  degree  in  y  ?  Answer  the  same  questions  for  the  other 
terms  of  the  multiplicand. 

5.  How  is  the  degree  of  an  integral  polynomial  determined  ? 
Give  three  illustrations  from  the  exercises  on  p.  42. 

6.  Name  the  degree  as  regards  x  of  each  polynomial  in  Exs. 
11  and  12  below. 

7.  Arrange  the  expression 

3ay^f  +  xf>-S  xY  —  6  T^y^  +  x^y  —  Sf-{-  5  y^ 
according  to   ascending  powers  of  y.     How  is  it  then  arranged 
with  reference  to  aj?     Of  what  degree  is  this  expression?     What 
is  the  degree  of  3  x^y^  ? 

Multiply : 

8.  6x^-2  +  5x  +  Si^hj  x^-\-5-x. 

[In  Exs.  8-16  arrange  both  multipUer  and  multiplicand  according  to  some 
letter  contained  in  each,  and  observe  that  the  product  has  then  a  correspond- 
ing arrangement.] 

9.  2a  +  a3-a2_iby  4-a2-fa. 

10.  Sa-x-4.ax^-{-x''-a^hj  a^-ax-^a^. 

11.  3xy'^-f-Sx^y-^x''hy-2xy-\-a^-^y\ 

12.  x-y'^  —  xy^  -f  y^  —  x^y-\-x^  by  x^-\-xy  —  y^. 

13.  4.hh-hr^-h^-\-27^\)yh-2r. 


35-37]  DIVISION  OF  ALGEBRAIC  EXPRESISIONS  45 

14.  Q>y^  +  ^ xY  +  2x'-3o?y-xfhy  y''  +  Zx'-2xy. 

15.  af  —  5  aj^2/^  —^f  —  &xy^-\- 15  ^y^  +  2  x^y  hj  'd  y"^  -{- x^  —  2  xy. 

16.  4.f-lQ>s^t^-^&^-\-^sH  +  ^st^hj^s'-f-^sH. 

17.  a"+i  -  3  a"+-  +  a  "+^  -  2  «"+'*  by  2  a"-i  +  3  a"-^  -  4  a'*-=^. 

18.  x^'y^  +  i»"+^?/  +  0^**+^  —  £c"+^?/2  by  or^  +  2/^  —  ic?/^  —  x^y. 

19.  In  Ex.  8,  of  what  degree  is  the  multiplicand  ?  the  multi- 
plier? the  product?  The  term  of  highest  degree  in  the  product 
is  the  product  of  what  two  terms  ? 

20.  Of  what  degree  is  the  multiplicand  in  Ex.  11  ?  the  multi- 
plier? What,  then,  should  be  the  degree  of  the  product?  Should 
all  the  terms  of  the  product  be  of  the  same  degree  ?     Why  ? 

II.     DIVISION 

36.  Law  of  exponents  in  division.  Since  division  is  the 
inverse  of  multiplication  (§  19),  therefore  the  results  of  §  30 
may  be  employed  to  find  the  law  of  exponents  in  division. 

Thus  :  since  a^  -  a^—  a^  therefore  a^  -j-  a^  =  ?  Is  o^  -r- oc^ 
equal  to  x^^  i.e.,  to  a:;^^?     Why? 

.  Write  the  following  indicated  quotients  and  explain  your 
answer  in  each  case  :  aJ  -^a^\  8^ -i-  s^;  2^"  -^  2^ ;  x^  -i-  x. 

How  is  the  exponent  of  the  product  of  two  powers  of  any 
given  number  obtained  (cf .  §  30)  ?  How,  then,  should  the 
exponent  of  the  quotient  be  obtained  ? 

If  m  and  n  are  positive  integers,  m  greater  than  /t,  and  x 
any  number  whatever,  then  (cf.  §  9,  also  Exs.  18,  20,  p.  39) 
the  above  results  may  be  expressed  in  symbols  thus : 

j,m  _j^  w-re  __  j,m — n 

This  equation  states  the  law  of  exponents  in  division  ; 
translate  this  law  into  common  language  (cf.  §  30). 

37.  Division  of  monomials.  Since  the  quotient  multiplied 
by  the  divisor  always  equals  the  dividend  (§8),  therefore 
12  a^-f-  3  a:^  =  ?  that  is,  what  is  the  number  which,  when 
multiplied  by  3  x'^.,  gives  12  a^  as  product  (gL  §  31)  ? 


46  HIGH  SCHOOL  ALGEBliA  [Ch.  IV 

Similarly:  8  a^rr^  ^  4  aV  =  ?  Why?  24mV-^8mY  =  ? 
Why?    -18  a^^"-^  6^362=?    Why?    Sml^xy^ -i- (^-^.m^xy)  ^^^ 

How  is  the  sign  of  the  quotient  determined  ?  the  coeffi- 
cient? the  exponents?  How  may  §  8  be  used 'to  test  the 
correctness  of  the  quotient  ?  From  the  above  write  a  rule 
for  dividing  one  monomial  by  another,  mentioning  the  sign, 
coefficient,  letters,  and  exponents  of  the  quotient  (cf.  §  31). 

EXERCISE  XXIV 

Perform  the  following  divisions ;  check  results  by  §  8 : 

1.  Q>a^^2a.  4.    - 1^  a^h' ^  (S  ah\ 

2.  15aV-f;3ax2.  5.   10  c«dV -- 5  c^de. 

3.  12  mV  -T-4:X^.  6.    —  45  m^n^  -^  ( —  ^  m^n). 

^      -48aV  ^^      -lmh\  ^^      3.1  (xyz)' 

12  aV   '  '     --i-mV'  '     -.Sa^yV' 

g   *15  7ty  ^2     -<33mny  ^^      - 12  g^^c^^ 


-mfgY\          13     _J/^.  ,„      2^ 


,»»+3 


25 //p2  -T\fh'  6x^ 

10.     — :r—rr'  14. 18.     . 

—  7  a^6  4^  m"?^'^  2  a;" 

19.  If  two  monomials  have  like  signs,  what  is  the  sign  of  their 
product  ?  of  their  quotient  ?  How  do  we  find  the  exponent  of 
any  given  letter  in  the  quotient  of  two  monomials  ? 

In  Exs.  20-25,  multiply  the  first  monomial  by  the  second; 
also  divide  the  second  monomial  by  the  first : 

20.  -  16  «^  51  m''.  23.   I  c  (m  +  n),  - 10  c'  (771  +  71)1 

21.  42  (p  +  qy,  —  14  (p  +  qy.  24.    8  ax^Y'"+%  —  2  ^^^+2^2  (m+»)^ 

22.  5afy%  15  a^^+y.  25.   13  (.t-:^)^,  -26(x-zy. 

38.  Division  of  a  polynomial  by  a  monomial.     Since  (§  32) 
a  (m  +  2;  —  ^)  =  «m  +  ax  —  ay, 
therefore         (am  +  t«:r  —  ay}  -i-  a  =  m  +  x  —  y, 
whatever  the  numbers  represented  by  a,  w,  a:  and  y.     Hence, 


;}7-;]8]  niVTSlON  OF  ALGKliUAIC  EXPRESSIONS  47 

To  divide  a  polynomial  hy  a  monomial^  divide  each  term  of 
the  polynomial    hy   the    monomial^   and  add  the  quotients  so 
obtained. 
E.g.,    (15  aV  -  10  hx^y  +  cV)  -^  5  x^  =^  S  a^x  -  2  bx^y  +  i  c\ 

EXERCISE  XXV 

Perform  the  following  indicated  divisions : 
,      4a»-12a^  ^     9  mhi^  + 12  mn^  -  30  m^n* 

D. 


4  a^  —  3  mn^ 

-2Axy  +  lSiK^y\  ^      -18a^-81a;  +  9a^ 

0  xy 


c2 


14  7-2- 

-3^0^ 
2l7-2s^  +  8 

-39> 

26  a^m' 

7  r 
^-52  aV- 

«%® 

1  a'W- 

13  a-m^ 

-4a«6^  +  6 

a^«Z>^ 

^     3  m  -  2  n  + 11  a;  ^ 

9mV  +  12myi2-30mV        ^ 

3m7i2  '  •  -2a26« 

11.  How  may  any  polynomial  whatever  be  divided  by  a  mono- 
mial ?  How  are  the  signs  of  the  several  quotient  terms  deter- 
mined ?     their  coefficients  ?     their  letters  ?     their  exponents  ? 

Divide  [and  check  the  work  in  each  case  (cf .  Ex.  10,  p.  2Q>)'] : 

12.  07-1-4  ax^  —  3  m^x  —  6  a7nx  by  —  x. 

13.  a^6%i3  -  4  a36V«  + 12  ci'ft'ic  by  4  rt^^s^ 

14.  1 7-^5  +  J  cr^^s^  —  I  r^s^  by  2  rs ;  also  by  f  rs. 

15.  a"*  —  2  a^'^^  -  5  a"*+2  +  9  a"»+^  by  a"* ;  also  by  a\ 

16.  2«+*  —  3  z""^  -h  4  aV  —  2;^  by  —  1 2:1 

17.  -  10  (h  -  1)*-'  -  6  (/i  -  1)^A:  + 15  (7i  -  1)^A:2  by  -  5  (h  - 1). 

18.  x(x-\-  yy  —^(x-\-  yf  +  x'^  (x  +yy  by  —  a?  (aj  +  ?/)l 

19.  2  (s  -  ^)-  -  s^  (s  -  O'^+i  -  5  (s  -  ^)-+3  by  i  (s  -  O'"-^ 
Separate  each  of  the  following  expressions  into  two  factors, 

one  of  which  is  a^ : 

20.  c^a^-hd^^.  22.    -Sx^y-{-5x^z-7x^. 

21.  aV  —  a^x^  +  aj2.  23.    —x^  +  Q  eV  -f  — -  • 

4 


48  HIGH  SCHOOL  ALGEBRA  [Ch.  IV 

In  Exs.  24-26,  group  the  like  powers  of  y  (cf.  Exs.  20-23)  : 

24.  ty^  -f  c?/'  -  ry'  -~3sy^-{-  y\ 

25.  ay*  --2by  —  S  cy^  —  my*  +  dy^  —  9  y. 

26.  (a  +  l)/-(«-l).v'  +  /-3  2/^-(3a4-4)2/3  +  a/. 

39.  Division  of  a  polynomial  by  a  polynomial.  Since,  by 
§  35,  the  product  of  (4:a^  +  2>x-2^  and  (jx^ -2x^+1  x^b) 
is  4  a:^  —  5  a^  +  20  a;^  +  45  a:2  +  a;  —  10,  therefore,  with  this  last 
expression  as  dividend,  and  a^— 2a:^+7a7  +  5as  divisor,  the 
quotient  must  be  4:x^-\-  S  x  —  2;  i.e., 
( 4  rcs  _  5  ^4  +  2 0  :z^  +  4  5  2^  +  a;  -  1 0 ) -f- (2^3  _  2  2^  +  7  rr  +  5) 

=  ^x^  +  Sx-2, 

The  process  of  obtaining  this  quotient  from  the  given 
dividend  and  divisor  will  now  be  explained. 

Since  the  dividend  is  the  product  of  the  divisor  by  the 
quotient,  therefore  the  highest  term  in  the  dividend  is  the 
product  of  the  highest  term  in  the  divisor  multiplied  by 
the  highest  term  in  the  quotient  (cf .  Ex.  19,  p.  45)  ;  and 
therefore  if  4  x^,  the  highest  term  in  the  dividend,  is  divided 
by  a^,  the  highest  term  in  the  divisor,  the  result,  4a^,  will  be 
the  highest  term  in  the  quotient. 

Moreover,  since  the  dividend  is  the  algebraic  sum  of  the 
several  products  obtained  by  multiplying  the  divisor  by  each 
term  of  the  quotient,  therefore  if  4  a;^  —  8  ic*  +  28  a^  4-  20  a^, 
the  product  of  the  divisor  by  the  highest  term  of  the 
quotient,  is  subtracted  from  the  dividend,  the  remainder, 
viz.,  3a::*  —  8a^  +  25a:2^^_;1^0^  ^[n  }yQ  ^\^q  g^^  Qf  ^\^q  prod- 
ucts obtained  when  the  divisor  is  multiplied  by  each  of  the 
other  terms  of  the  quotient  except  this  one. 

For  the  reason  given  above,  if  3  a;*,  the  highest  term  of  this 
remainder,  is  divided  by  a^,  the  highest  term  of  the  divisor, 
the  result,  3  x,  is  the  next  highest  term  of  the  quotient. 

By  continuing  this  process  all  the  terms  of  the  quotient 
may  be  found.     The  work  may  be  arranged  as  follows  : 


J9]  DIVISION   OF  ALGEBRAIC  EXPRESSIONS  49 


DIVIDEND 


x3-2a:2+7x+5 


4x2  +  3x-2 


(x3-2x2  +  7x+6).4x2=       4  x5-8a;H  28x3+20x2 

3x4-  8x3  +  25x2+ x-10 — -' 

(a;3_2a;2+7x  +  5).3x  =  Sx^-  6x3+21x2+ 15 x  Quotient 

-  2x3+  4x2-14  X- 10 
(x3-2x2+7x+5)  .(-2)=  -  2x3+  4x2-14 x- 10 

0 
Check 

When  X  =  1,  dividend  =  55,  divisor  =  11,  and  quotient  =  5,  as  it  should. 

Even  if  it  is  not  known  beforehand  that  the  dividend  is 
the  product  of  two  polynomials,  the  process  of  division  may 
still  be  applied  as  above.  This  process  may  be  formulated 
thus  : 

(1)  Arrange  both  dividend  and  divisor  according  to  the 
descending  powers  of  some  one  of  the  letters  involved  iii  each, 
and  place  the  divisor  at  the  right  of  the  dividend. 

(2)  Divide  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor y  and  write  the  result  as  the  first  term  of  the  quotient. 

(3)  Multiply  the  entire  divisor  by  this  first  quotient  term^ 
and  subtract  the  result  from  the  dividend, 

(4)  Treat  the  remainder  as  a  new  dividend^  arranging  as 
before^  and  repeat  this  process  until  a  zero  remainder  is  reached^ 
or  until  the  remainder  is  of  lower  degree  in  the  letter  of  arrange- 
ment than  the  divisor. 

EXERCISE  XXVI 

Divide  (and  check  your  results  by  §  25)  : 

1.  ar^+7a;  +  12by  ic  +  3.  5.    IS  x +6x^-i-6  by  3x-{-2. 

2.  ay^-x-20hj  x-5.  6.   8  +  3  a^-  14  a;  by  2-3a;. 

3.  b'-6b-16hjb  +  2.  7.    10a^+lla2_8byl-2al 

4.  s2-14s  +  49by  s-7.  8.   Sx^ -4.a^-7  hj  -  sc^-l. 

9.    c3  +  6c24.12c+8by  c+2. 
10.   2a^-{-llx''-\-19x-\-10hy2x''-i-7xi-5. 
U.   75  m-  +  m^  - 15  m'  -125  by  25-\-m'-  10  ml 


50  HIGB  SCHOOL  ALGEBRA  [Ch.  IV 

12.  p*-\-4:p^-\-6p''  +  5p  +  2hjp'+p  +  l. 

13.  2ic^-f6a^  —  4a;  —  5a^  +  lbya^  —  a;  +  l. 

14.  3a^+3a^  +  3  +  3a  +  a^  +  5a3by  1  +  a. 

[Here,  as  in  arithmetical  "long  division,"  labor  may  be  saved  by  "bring- 
ing down  "  at  any  stage  of  the  vs^ork  only  so  much  of  the  remainder  as  is 
needed  for  the  next  step.] 

15.  Divide  6  a^o^  -  4  a^x-4:ac(f  +  a"^ -\- x'^  hj  a^ -\- x^  -  2  ax. 

SOLUTION 

X*  -iax^  +  6  a^x^  -  4  a%  +  a*  bc^  -  2  ax  +  a^ 

x^  —  2  ax^  +  (i^x^ ^•■^  —  2  ax  -{■  a^ 

—  2  ax^  +  5  a'^ic-^  —  4  a^x 
-2ax^  +  4  a^x^  -  2  a^x 

a-x-  —  2  a^x  +  a* 
a-x^  -2a^x-\-  g^ 
0 
Note.  To  make  the  explanation  of  §  39  apply  when  two  or  more  letters 
are  involved,  replace  "highest  term"  by  "term  of  highest  degree   in  the 
letter  of  arrangement." 

16.  In  Ex.  15  perforin  the  division  when  both  dividend  and 
divisor  are  arranged  according  to  the  descending  powers  of  a. 

17.  Divide  4  a^i/^  +  8  a^  +  2/^  +  8  ic^2/  ^J  ?/  +  ^  x. 

18.  Divide  2  a^  +  A;^  -  5  o^k  -  4  ak^  +  6  a'k'  by  ¥  +  a^-  ak. 

19.  (10  x^y^  -^  ii^  -10  xhf  -\-5  xy'  -  5  x'y  -f)-^(a^+y^-2 xy)  =  ? 

20.  If  the  partial  quotient,  at  any  stage  of  the  process  of  divi- 
sion, is  multiplied  by  the  divisor,  and  the  corresponding  remain- 
der added,  how  must  the  result  compare  with  the  dividend  ? 

21.  What  check  for  division  is  suggested  by  Ex.  20  ?  Is  this 
check  more  or  less  complete  than  that  given  in  §  25  ?     Explain. 

22.  Divide   2a^  +  a;^  +  49a^-13aj-12by  a^-2«2  4.7aj4_3. 

[Since  there  is  no  term  in  x^  in  the  dividend,  care  must  be  used  to  keep 
the  remainders  properly  arranged.] 

Divide  (and  check  the  results  as  the  teacher  directs) : 

23.  v^  —  'v*  —  l-}-2v-\-v^  —  v-hjv  —  l-\-  v\ 

24.  0(^^-41  a-120  by  a2  +  4a-|-5. 


39]  DIVISION  OF  ALGEBRAIC  EXPliEbSIONS  51 

25.  m*  +  16  +  4  m^  by  2  m  +  m^  +  4. 

26.  T\x*-ia^'y  +  lixY  +  ixfhy^x-\-iy. 

27.  1.2  aa;^  -\-a^x^-2a'-  3.4  a-V  +  6  aa;  by  6  aa;  -2  al 

28.  {2x-{-3a^-l+2x^){l-\-a^-x)hjl  +  x  +  x^. 

29.  a^  -  6^  by  (a^  +  63^)  (a  +  6)  +  a'b\ 

30.  a^  4-  6^  +  c^  —  3  a6c  by  a^  -{- b^ -{- c^  —  ab  —  ac  —  be. 

31.  a;*-3a^  +  a^  +  2a;-l  byaj2-aj-2. 

fin  Ex.  31  the  complete  quotient  is  ic'^  -  2  x  +  1  +     ~  ^  '^       .~| 

32.  v^  —  v-{-7hjv-\-A.  34.   2  s^  —  3  s  +  8  by  s^  —  4. 

33.  a^  — 1  by  a  +  1.  35.   a^  +  a;  —  25  by  a;  —  3. 

36.  a*-7a2-9a-6a3-6by3+a2-2a. 

37.  3  a;^  4- 11  a^  +  11  a;^  +  9  a;  +  10  by  4  x  +  5  4-  a^. 

38.  Divide  p^ -{-  q^  by  p -^  q  until  4  quotient  terms  are  obtained  ; 
divide  1  by  1  —  r  to  8  quotient  terms ;  1  by  1  —  mx  to  4  quotient 
terms ;  and  a  by  a  —  a;  to  5  quotient  terms. 

Divide : 

39.  cd-d2  +  2c2by  c  +  c?.  42.  h^ - 7i^  by  h^  +  Ic^. 

40.  oc^  —  i/  by  x  —  y.  43.  a-"  —  a;-"  by  a"  —  a;". 

41.  a*-166^by  a-26.  44.  u^'' -^  11  u"" -{- 30  by  u"" -\- 6. 

45.  a5'"+'*  —  a?**?/""^  —  a:*"?/** +2/^""^  t>y  ^"^  ~  2/**"^- 

46.  af"+"-^  -  3  xY"  "^  —  5  a;"*"  V  + 1^  /''-^  by  a;"  -  5  y. 

47.  Divide  a6c  +  aoif  +  a^  +  a6a;  +  6a;^  +  ca;^  +  aca;  +  bcx  by 
a^  -\-  ax -\-  ab  -\-  bx. 


Solution.     Since  x  occurs  in  more  terms  than  any  other  letter,  it  will 
be  best  to  arrange  the  work  thus  (cf.  Exs.  24-26,  p.  48)  : 

or^  +  (a  +  ?>  +c)  x2  +  (ab  -\-  ac  ■{-  bc)x  +  abc  \c^  +  (a  +  b)x  +  ab 
x^  +  (a  +  h)x/^        +  ahx 

cx^  +  {ac  +  bc)x  +  ahc 

cx/^  +  {ac  +  hc)x  +  cthc 


0  I 


62  HIGH  SCHOOL  ALGEBRA  [Ch.  IV 

Divide : 

48.  1/ -\- cy -\- cd -\-  dy  by  2/  +  c. 

49.  -ab-\-ay  +  y^-byhy  y-b. 

50.  f-\-2  dy'  +  2  /  +  ^^2/  +  4  d?/  +  2  d-  by  /  +  2  ^2/  +  d^- 

51.  a-f{a-\-&)+^y'-y^-2,ay'  +  aY-2ayhySf-y  +  a. 

52.  Divide  a^  —  21  by  x  —  a)  note  that  the  remainder  is  what 
the  dividend  would  become  if  a  were  substituted  for  x. 

53.  Divide  a^  +  Sa^  +  l  by  x  —  a]  note  that  the  remainder 
differs  from  the  dividend  only  in  that  a  replaces  x. 

54.  Divide  m^4-7  by  m  —  c  and  compare  the  remainder  with 
the  dividend.  Similarly,  divide  v^  —  1  by  v  —  2 ;  5  m^  —  8  m  +  3 
by  m-3;  2/^-4/+ 32/-1  by  y-h-,  2?-* -7^  + 10  by  r-1; 
by  r  —  2 ;  by  r  —  3. 

55.  Divide  2  xy^  +  Sx*  —  4:  x^y^  —  7a^y-\-y^  by  a^-\-y^  —  xy,'  ar- 
ranging first  according  to  powers  of  x,  then  according  to  powers 
of  y,  and  compare  the  results. 

56.  As  has  just  been  seen  in  Ex.  55,  the  form  of  the  quotient 
depends  upon  the  choice  of  the  letter  of  arrangement  when  the 
division  is  not  exact ;  is  this  the  case  when  the  division  is  exact  ? 

40.  Finite  numbers.  As  we  pass  from  left  to  right  the 
numbers  of  the  series  2,  2^,  2^  2*,  etc.,  increase  without  end ; 
and  the  numbers  of  the  series  1,  J,  |^,  etc.,  decrease  without 
end.  Hence  we  see  that,  in  mathematical  operations,  there 
may  arise  numbers  which  are  greater,  and  others  which  are 
smaller,  than  any  fixed  number  that  we  can  name  or  even 
conceive  of;  such  numbers  are  called  infinitely  large  and  in- 
finitely small  numbers,  respectively.  All  other  numbers  are 
called  finite  numbers.  An  infinitely  large  number  is  repre- 
sented by  the  symbol  00. 

41.  Zero.  Operations  involving  zero,  (i)  The  result  of 
subtracting  any  given  finite  number  from  itself  is  called 
zero  (cf.  §  13).  Thus  if  a  represents  any  finite  number, 
then        I  a  —  a  ==  0. 


39-41]  DIVISION   OF  ALGEBBAIC  EXPRESSIONS  53 

(ii)     From  this  definition  of  zero  and  the  definitions  of 
addition,  subtraction,  etc.,  already  given,  it  follows  that,  if  k 
is  any  finite  number  whatever, 
then  k-\-0  =  k-0  =  k, 

and  A;  .  0  =  0  .  ^  =  0. 

J£^.^.,  ^  .  0  =  0  because  k  •  0  =  k  •  {a  —  a)  —  ka  —  ka  =  0. 

(iii)     If  k  is  any  finite  number  whatever,  then 

A;  -7-  0  =  no  finite  number  whatever. 
For,  if  A; -7-0=/,  a  finite  number,  then/-  0  would  equal  k 
(§  19),  but  this  is  impossible  (ii). 

(iv)  0-^0  =/  [i^iiy  finite  number 

for  /•0  =  0. 

(v)  From  (iii)  and  (iv)  above  it  follows  that  we  must 
not  divide  hy  zero^  since  doing  so  leads,  at  best,  to  an  inde- 
terminate result. 

EXERCISE  XXVII 

1.  When  the  values  1,  \,  \,  \,  y^g,  •  •  •  are  assigned  to  x,  how 
do  the  successive  values  of  the  fraction  5/a;  compare  ?  Can  you 
name  a  number  so  large  that  none  of  these  values  will  exceed  it? 
Can  you  name  a  number  so  near  0  that  none  of  the  series  of  num- 
bers 1,  -^,  \,  ^,  yi^,  •   •  •  will  be  still  nearer  to  0  ? 

2.  What  is  meant  by  an  infinitely  small  number  ?  by  an  in- 
finitely large  number  ? 

3.  Define  zero.     How  does  it  follow  from  your  definition  that 

(1)  3-0  =  3?  (2)   0-5  =  0? 

4.  Can  the  equation  ax  =  0  be' true  if  neither  a  nor  x  is  zero? 
Does  it  require  that  both  a  and  x  should  be  zero  ? 

5.  What  is  the  value  of  f  ?  Why  ?  What  is  the  value  of  0/a, 
wh^re  a  is  any  number  except  zero  ? 

6.  May  ^  =  5?  1000?  -72?  ^?  Explain.  What  is 
meant  by  saying  that  ^  gives  an  indeterminate  quotient  ? 

7.  The  quotient  ^  cannot  be  a  finite  number.  W^hy  ?  Will 
it  be  an  infinitely  large  or  an  infinitely  small  number  (cf.  Ex.  1)? 


54  HIGH  SCHOOL  ALGEBllA  [Ch.  IV 

42  *  Some  elementary  laws.  What  is  the  meaning  of  the  expres- 
sion 5  +  2  +  8  (cf .  §  §  4  and  10)  ?  of  2  +5  +  8  ?  Wherein  do  these 
expressions  differ? 

(i)  Although  a  change   in  the  order  in  which   operations   are 
performed   may,  in   general,   change   the   result    (cf.  §  10),  yet 
some  such  changes  of  order  do  not  affect  the  result.     Thus : 
5  +  2  +  8=2  +  5  +  8  =  8  +  5  +  2, 

5.2.8  =  2.5.8  =  8.5.2, 
5 +  2  +  8  =  5  + (2 +  8)  =  5  +  10, 
5.2.8  =  5. (2. 8)  =  5. 16, 
and  5.(2  +  8)  =  5.2  +  5.8. 

(ii)  Moreover,  based  upon  our  experience  with  particular  sets 
of  numbers,  we  have  silently  assumed,  in  the  preceding  pages, 
that  the  above  changes  may  be  made  with  any  numbers  whatever 
without  affecting  the  result.  Thus,  if  a,  6,  and  c  represent  any 
numbers  whatever  (positive,  negative,  integral,  etc.),  we  have  as- 
sumed (without  proof)  that : 

a  +  &  +  c  =  6+a  +  c  =  c  +  a  +  6,  etc.,  (1) 

a  'h  '  c  =  h  '  a  •  c  =  c  •  a  'h,  etc.,  (2) 

a^h  +  c  =  a-\-(h  +  c),  etc.,  (3) 

a  'b  '  c  =  a  '  (b  '  c),  etc.,  (4) 

and  a-  {b  +  c)  =  ab-^ ac.  (5) 

Of  these  equations,  (1)  states  what  is  known  as  the  commutative 
law  of  addition;  (2),  the  commutative  law  of  multiplication;  (3), 
the  associative  law  of  addition;  (4),  the  associative  laiv  of  multipli- 
cation; and  (5),  the  distributive  law  of  multiplication  as  to  addi- 
tion: all  of  them  taken  together  are  often  spoken  of  as  the 
combinatory  laws  of  algebra. 

These  laws  are  easily  verified  in  any  particular  cases :  through- 
out this  book  we  shall  continue  to  assume  their  correctness.  We 
wish,  however,  to  point  out  to  the  pupil  that  mere  verifications, 
however  numerous,  do  not  establish  a  general  law. 

*  This  article  may,  if  the  teacher  prefers,  be  omitted  till  the  subject  is 
reviewed.    For  a  full  discussion  of  these  laws  see  El.  Alg.  Chap.  IV. 


CHAPTER   V 
EQUATIONS  AND   PROBLEMS 

43.  Equation.  Members  of  an  equation.  A  statement 
that  each  of  two  expressions  has  the  same  value  (i.e.,  repre- 
sents the  same  number)  as  the  other,  is  called  an  equation. 
These  two  expressions  are  called  the  members  of  the  equa- 
tion, the  expression  preceding  the  sign  of  equality  being  the 
first  member,  and  the  other  the  second  member. 

Thus,  Sx—16  =  Sx-i-4:  is  an  equation ;  8  a;  —  16  is  its 
first  member,  and  S  x -\- 4:  its  second  member. 

Remark.  In  algebraic  work,  the  equation  is  a  most  important 
instrument ;  to  it  is  due  the  chief  advantage  of  algebra  over  arith- 
metic. We  have  already  seen  some  evidence  of  this  in  §  3,  but 
much  more  is  to  follow.  In  a  recent  book  Sir  Oliver  Lodge  says  : 
"  An  equation  is  the  most  serious  and  important  thing  in  mathe- 
matics." 

44.  Conditional   equations.      Identical   equations.      Is  the 

statement  Sx  —  lQ=Sx-\-4:,  true  when  x  =  l?  when  x=2? 
when  a:  =  3  ?  when  a^  =  4  ?  when  x  =  5?  Answer  the  same 
questions  with  regard  to  2x=  (x  -{- 1)^  —  Qa^  +  1). 

The  equation  Sm-\-  ^n  =  22  is  true  if  m  =  4  and  n  =  2, 
but  is  not  true  for  any  other  positive  integral  values  of  m 
and  n ;  while  the  equation  Sx^  +  k  —  a^  =  k-\-2x^is  true  for 
all  values  that  may  be  assigned  to  x  and  k. 

An  equation  which  is  true  for  all  values  that  may  be  as- 
signed to  its  letters  is  called  an  identical  equation,  and  also 
an  identity;  while  one  which  is  true  only  on  condition  that 
certain  particular  values  be  assigned  to  its  letters,  is  called 
a  conditional  equation.  In  the  following  pages  we  shall  use 
the  word  equation  to  mean  conditional  equation  unless  the 
contrary  is  expressly  stated. 

55 


56  HIGH  SCHOOL  ALGEBRA  [Ch.  V 

As  we  shall  see  later,  by  performing  the  indicated  opera- 
tions the  two  members  of  an  identity  may  be  reduced  to 
exactly  the  same  form;  hence  the  name  "identical  equation." 

45.  Roots  of   an   equation.     Checking.     The   roots   of   an 

equation  are  those  values  which  satisfy  the  equation;  ^.e., 
they  are  those  values  which,  when  substituted  for  the  letters 
the  equation  contains,  make  the  two  members  identical. 

Any  process  by  which  the  roots  of  an  equation  are  found 
is  called  solving  the  equation. 

The  final  test  of  the  correctness  of  supposed  roots  is  to 
substitute  them  for  the  letters  in  the  equation ;  if  they  satisfy 
the  equation  they  are  roots,  otherwise  not.  This  process  is 
called  checking  the  roots.     Thus,  4  is  a  root  of  the  equation 

8a;-16  =  3a:H-4, 
because  8-4  —  16  =  3-4+4.         [each  member  being  16 

46.  Some  axioms  and  their  uses.  The  following  principles, 
usually  called  axioms^  are  useful  in  solving  equations : 

1.  If  equals  (i.e.,  equal  numbers^  are  added  to  equals^  the 
sums  are  equal. 

2.  If  equals  are  subtracted  from  equals,  the  remainders  are 
equal. 

3.  If  equals  are  multiplied  hi/  equals^  the  products  are  equal. 

4.  If  equals  are  divided  hy  equals,  the  quotients  are  equal. 
Here,  however,  as  elsewhere,  it  is  not  permissible  to  divide 
by  zero  [cf.  §  41  (v)]. 

The  correctness  of  these  axioms  rests  upon  the  fact  that 
equal  numbers  are  in  reality  the  same  number,  differing  at 
most  in  form.  Thus,  24  +  11,  7  •  5,  and  6^  —  1  are  merely 
different  forms  of  writing  35. 

Suggestion  to  the  Teacher.  It  is  strongly  recommended  that  the  teacher 
illustrate  the  physical  meaning  of  an  equation,  and  also  the  meaning  of  the 
above  axioms,  by  means  of  a  pair  of  balances  (easily  made,  if  not  provided 
by  the  school). 


44-47]  EQUATIONS  AND   PROBLEMS  57 

47.  Solution  of  equations.  To  show  how  the  above  axioms 
may  be  used  in  solving  equations,  let  it  be  required  to 
solve  the  equation  H  x—lQ  =  '^  x  -\-  4,  ^.e.,  to  find  the  value  of 
X  which  satisfies  it. 


SOLUTION 

Since 

Sx-16  =  3x-\-4r, 

therefore 

Sx 

-16  +  16  =  3x  +  4+16, 

[Axiom  1 

i.e., 

8  a^  =  3  a;  +  20, 

and  therefore 

8x-3^  =  3a;  +  20-3a-, 

[Ax.  2 

i.e., 

5x  =  20, 

whence 

x  =  4.. 

[Ax.  4 

CHECK 

On  substituting  4  for  x  in  the  original  equation,  that  equation 
becomes 

8. 4- 16  =3. 4  + 4,  i.e.,  16  =  16  ; 

hence  the  equation  is  satisfied,  and  4  is  a  root  (cf.  §  45). 

EXERCISE  XXVIII 

1.  Is  2  a  root  of  a^  — 5  07  +  6  =  0  ?    Is  3  also  a  root  ?    Explain. 
How  may  we  check  a  supposed  root  of  an  equation  ? 

Solve  the  following  equations,  give  the  reasons  for  each  step  in 
the  work,  and  check  the  roots : 

2.  10a;  =  40.  10.  3m  +  2  =  m4-30. 

3.  Sy  =  -32.  11.   7a;-10  =  5x  +  18. 

4.  k  +  l  =  7.  12.-4^  =  3  +  ^  —  15. 

5.  m-9  =  4.  13.  20-12w  +  5  =  0. 

6.  2i;  +  7  =  63.  '  14.  13s-9-2s  =  24. 

7.  2v-7  =  63.  15.  7x-55  =  lS-2x-l. 

8.  46  =  5s-4.  16.   6v-(v-3)-12  =  0. 

9.  -13  =  3a;  +  8.  17.  2/'-(/+2/  +  8)  =  -6. 

18.  In  Exs.  7-13,  point  out  the  members  of  each  equation. 
Which  is  the  first  member  of  the  equation  in  Ex.  14  ?  What  is 
the  other  member  called  ? 


58  BIGII  SCHOOL  ALGEBRA  [Ch.  V 

19.  What  is  meant  by  solving  an  equation  ?  Describe  briefly 
the  process  used  in  solving  an  equation. 

20.  Are  the  equations  in  Exs.  2-17,  above,  conditional  equations 
or  identities  ?   Why  ?   In  which  class  of  equations  would  you  place 

2x  +  S  =  2(4:X-{-3)-(6x  +  3)?     Why  ? 

21.  If  2  a  is  subtracted  from  each  member  of  the  equation 
5x-\-2a  =  3x-\-4:b,  what  is  the  resulting  equation  ?  What  does 
this  show  with  reference  to  removing  a  term  from  the  first  to  the 
second  member  of  an  equation  ?  Is  the  same  thing  true  when  a 
term  is  removed  from  the  second  member  to  the  first?  Show 
this  by  adding  —3  a;  to  each  member  of  the  given  equation. 

48.  Transposition.  Directions  for  solving  equations.  Re- 
moving a  term  from  one  member  of  an  equation  to  the  other 
is  called  transposing  that  term. 


If 

x-}-a  =  b, 

then 

x-\-a  —  a  =  b- 

-a. 

[Ax.  2 

i.e., 

x=h- 

-a. 

[•.•a-a  =  0 

Again,  if 

x  =  b- 

-a, 

then 

x-\-a  =  b- 

-a-\-a, 

[Ax.  1 

i.e., 

x-^a  =  b. 

[...  -a-i-a  =  0 

Hence,  since  a  msij  represent  any  term  whatever,  a  term 
may  he  transposed  from  one  member  of  an  equation  to  the  other 
by  merely  reversing  its  sign  (cf.  also  Ex.  21,  above). 

For  solving  equations  such  as  those  considered  in  §  47  the 
following  directions  may  now  be  given:. 

1.  Transpose  all  the  terms  containing  the  unknown  number 
to  the  first  member  of  the  equation.,  and  all  other  terms  to  the 
second  member. 

2.  Unite  the  terms  of  each  member.,  and  then  divide  both 
members  by  the  coefficient  of  the  unknown  number. 

3.  Check  the  root  thus  found  by  substituting  it  in  the  given 
equation. 


47-48]  EQUATIONS   AND    PROBLEMS  69 

Ex.  1.   Solve  the  equation  4  s  —  15  =  2  s  + 11. 

SOLUTION 

Transposing,  we  have     4  s  —  2  .s  =  11  + 15; 
uniting  like  terms,  2  s  =  26; 

dividing  by  2,  s  =  13. 


Check:                     4-13-15  = 

:  2  •  13  + 11.               f  each  mem 

Lber  is  37 

EXERCISE  XXIX 

Solve  (and  check)  the  following  equations : 

2.  32/-5  =  22. 

3.  -10  =  6  a  +  8. 

13. 

^-7  =  12. 
4 

4.  3(aj-5)  =  48. 

14. 

2^  +  ^  =  ??. 

5.  iz-\-2-z^n. 

3      6 

6.  A.-x  =  -ll-\-2x. 

15. 

^     ^^^  =  m  +  10. 
3y-7  =  A-2y-5. 

7.    {d-\-lf-d''  =  -ll, 
a  20-5fc  =  3A;  +  3. 
9.   -8a;  =  4(aj-2)  +  10. 

16. 
17. 

10.  4(-3  +  /i')  =  (2/i-3)2. 

XX.  1^-1  =  10. 

18. 
19. 

i(2/-6)  =  K2/-2). 
x-{-l      x  +  6          o 
-    7     +     2     -      ^* 

[Multiply  both    members   of   the 

20. 

3ia  =  5a-9ia-16. 

equation  by  12  (see  Ax.  3).] 

21 

12-3a;  +  20  =  44  +  3a;. 

•  ^'-  f'-ro-''- 

22. 

x-9     x-5     ^ 
3     ~    12     ' 

23.  14A;-(20-7A:-2)=:6A:+68. 

24.  (c  +  5)(2c-l)-(2c-3)(c  +  7)=0. 

25.  What  is  meant  by  transposing  a  term  from  one  member  of 
an  equation  to  the  other  ?  What  change  must  be  made  in  a  term 
thus  transposed? 

26.  State  in  order  the  axioms  thus  far  used  in  solving  equa- 
tions. Illustrate  the  use  of  each.  Why  does  the  division  axiom 
not  apply  when  the  divisor  is  zero?    [Cf.  §  41  (v).] 

HIGH  SCH.  ALG.  —  5 


00  HIGH   SCHOOL   ALGEBRA  [Ch.  V 

27.   Point  out  the  fallacy  in  the  following  reasoning : 

If  X  =  a, 

then  x^  =  ax, 

and  xF  —  a^  =  ax  —  a',       [subtracting  a^  from 

each  member 
i.e.,  (x  +  a)(x  —  a)  =  a  (x  —  a) ; 

therefore  2a(x  —  a)=a(x  — a),  [since  x  =  a 

and,  therefore,  2  =  1.  [dividing  by  a  (x  —  a) 

49.  Translation  of  common  language  into  algebraic  lan- 
guage, and  vice  versa.  Tlie  equation  a^  —  8  =  3  is  an  algebraic 
sentence  ;  it  may  be  translated  into  common  (verbal)  lan- 
guage thus :  "  X  exceeds  8  by  3  "  or  ^^x  is  3  greater  than  8." 

Similarly,  the  verbal  statement  "  the  excess  of  a  over  the 
product  of  8  and  t  is  9,"  when  expressed  algebraically,  be- 
comes a  —  st=9. 

In  order  to  use  equations  easily  in  the  solution  of  prob- 
lems we  must  learn  to  translate  freely  from  either  of  these 
two  languages  into  the  other. 

EXERCISE   XXX 

Write  as  algebraic  sentences : 

1.  Nine  is  2  greater  than  x. 

2.  2/  is  8  less  than  3  x. 

.    3.   a^  exceeds  2  a  by  1.  • 

4.  The  excess  oi  Sx  over  6  a;  is  2  a?. 

5.  The  difference  of  two  given  numbers  is  five  less  than  three 
times  their  sum. 

[Hint.    Let  a  and  b  be  the  given  numbers.] 

6.  The  product  of  two  given  numbers  exceeds  half  the  larger 
number  by  17. 

7.  Twenty-one  is  divided  into  two  parts,  the  smaller  of  which 
is  p.  What  is  the  larger  part?  Express  by  an  equation  that 
the  larger  part  exceeds  the  smaller  by  3. 


48-49]  EQUATIONS  AND  PROBLEMS  61 

8.  Translate  into  verbal  language  the  equations  in  Exs.  5-11, 
p.  57.  In  how  many  different  ways  may  we  translate  the  equa- 
tion in  Ex.  8  ? 

9.  A  father  is  now  4  times  as  old  as  his  son.  Eepresent 
the  age  of  each  5  years  ago ;  5  years  hence.  Also  express  by  an 
equation  the  fact  that  5  years  ago  the  father's  age  was  7  times 
that  of  his  son. 

[Hint.     Let  the  son's  present  age  be  s  years.] 

10.  Translate  into  algebraic  language  the  following  statement : 
a  rectangular  flower  bed  whose  length  is  y  feet,  and  whose 
width  is  6  feet  less  than  its  length,  contains  40  square  feet. 

11.  If  butter  costs  m  cents  a  pound,  eggs  n  cents  a  dozen,  and 
milk  r  cents  a  quart,  express  in  algebraic  language  that 

(1)  the  combined  cost  of  8  qt.  of  milk  and  6  doz.  eggs  is  $  1.90. 

(2)  the  cost  of  9  qt.  of  milk  is  30  cents  less  than  the  cost  of 
2i  lb.  of  butter. 

12.  Express  as  common  fractions :  50  %  of  n  dollars  ;  26  %  of 
k  bushels ;  m  %  of  $  525.  Show  that  the  amount  of  x  dollars  at 
5  %  simple  interest  for  3  years  is  cc  -f-  -^^  x. 

13.  If  the  units'  digit  of  a  number  is  2,  the  tens'  digit  4,  the 
number  itself  is  4  •  10  +  2,  i.e.,  42.  What  is  the  number  whose 
units'  digit  is  8  and  whose  tens'  digit  is  3  ?  the  number  whose 
tens'  digit  is  x  and  whose  units'  digit  is  a;  4-  7  ? 

14.  The  smallest  of  three  consecutive  integers  is  a ;  what  are 
the  other  two  ?  If  n  is  any  integer,  2  n  is  an  even  integer ;  write 
the  even  integer  next  higher  than  «2  n ;  next  lower  than  2  n. 
Write  the  odd  integer  next  lower  than  2  n ;  next  higher  than  2  n. 

15.  A  walks  2i  miles  an  hour ;  B,  3  miles  an  hour.  How  far 
does  each  walk  in  3  hours  ?  in  f  hours  ?  How  much  farther 
than  A  does  B  walk  in  1  hour  ?  Express  by  an  equation  that  in 
t  -\-2  hours  B  walks  3  miles  farther  than  A. 

16.  At  the  rate  given  in  Ex.  15,  in  how  many  hours  will  A 
walk  10  miles  ?  15  miles  ?  s  miles  ?  Answer  the  same  ques- 
tions for  B. 


62  HIGH  SCHOOL  ALGEBRA  [Ch.  V 

17.  If  I  can  do  a  certain  piece  of  work  in  6  days,  what  part  of 
it  can  I  do  in  1  day  ?  in  5  days  ?  in  a?  days  ?  If  I  can  finish 
a  job  in  d  days,  what  part  can  I  finish  in  1  day  ?     in  3  days  ? 

50.  Problems  leading  to  equations.  A  problem  is  a  ques- 
tion proposed  for  solution;  it  always  asks  to  find  one  or 
more  numbers  which  at  the  beginning  are  unknown,  and  it 
states  certain  relations  (conditions)  between  these  numbers, 
by  means  of  which  their  values  may  be  determined. 

In  solving  a  problem  the  important  steps  are : 

1.  To  represent  one  of  the  unknown  numbers  involved  in  the 
problem  by  some  letter^  as  x. 

2.  To  translate  the  common  language  of  the  problem  into 
algebraic  language. 

3.  To  solve  the  equation  thus  founds  —  called  the  equation  of 
the  problem. 

4.  To  check  the  result. 

These  steps  are  illustrated  in  the  solutions  of  the  follow- 
ing problems. 

Prob.  1.  The  sum  of  the  ages  of  a  father  and  son  is  54  years, 
and  the  father  is  24  years  older  than  the  son.     How  old  is  each  ? 

Solution.     Stated  in  verbal  language,  the  given  conditions  are : 

(1)  The  number  of  years  in  the  father's  age  plus  the  number  of 
years  in  the  son's  age  is  54. 

(2)  The  number  of  years  in  the  son's  age  plus  24  equals  the 
number  of  years  in  the  father's  age. 

To  translate  these  conditions  into  algebraic  language, 
let    X  stand  for  the  number  of  years  in  the  son's  age ; 
then,  by  the  second  condition, 

0?  +  24  stands  for  the  number  of  years  in  the  father's  age, 
and,  by  the  first  condition, 

a?  +  cc  +  24  =  54, 
which  is  the  equation  of  the  problem. 

Solving  this  equation,  we  find  x  — 15,  whence  a;  +  24  =  39. 
On  substitution  in  the  problem,  these  numbers  are  found  to  satisfy 


49-50]  EQUATIONS  AND  PliOBLEMS  63 

its  conditions  (i.  e.,  to  check) ;  therefore  the  father's  and  son's 
ages  are,  respectively,  39  years  and  15  years. 

Prob.  2.  A  boy  was  given  39  cents  with  which  to  buy  3-cent 
and  5-cent  postage  stamps,  and  was  told  to  purchase  5  more  of 
the  former  than  of  the  latter.  How  many  of  each  kind  should 
he  purchase  ? 

Solution.     Stated  in  verbal  language,  the  given  conditions  are: 

(1)  The  total  expenditure  is  39  cents. 

(2)  There  are  to  be  5  more  3-cent  stamps  than  5-cent  stamps. 
To  translate  these  conditions  into  algebraic  language, 

let         X  stand  for  the  number  of  5-cent  stamps  purchased, 
then     6x  stands  for  the  number  of  cents  in  their  cost; 
and,  by  the  second  condition, 

a?  -f  5  stands  for  the  number  of  3-cent  stamps  purchased, 
and        3  07  4-15  stands  for  the  number  of  cents  in  their  cost; 
hence,  by  the  first  condition, 

5aj  +  3a;-f  15  =  39, 
which  is  the  equation  of  the  problem. 

Solving  this  equation,  we  have  cc  =  3,  whence  a;  +  5  =  8.  Sub- 
stitution in  the  problem  shows  that  .these  values  check.  Hence 
the  number  of  5-cent  stamps  is  3,  and  the  number  of  3-cent 
stamps  is  8. 

Prob.  3.  If  a  certain  number  is  diminished  by  6,  and  twice 
this  diiference  is  added  to  5  times  the  number,  the  result  will 
equal  88  minus  3  times  the  number.     What  is  the  number  ? 

Solution.     To  form  the  equation  of  the  problem, 
let  n  represent  the  number  sought, 

then         6  n  =  5  times  the  number, 

and  2  {n  —  6)  =  twice  the  difference  of  this  number  and  6, 

and  88  —  3  71  =  88  minus  3  times  the  number. 

Hence  the  given  condition  becomes 

5n-f  2(n-6)=88-3w. 

The  solution  of  the  equation  gives  n  =  10,  which  checks;  there- 
fore 10  is  the  required  number. 


64  nWU  SCHOOL   ALGEBRA  [Ch.  V 

Prob.  4.  A  number  consists  of  two  digits  whose  sum  is  5 ;  if 
the  digits  are  interchanged,  the  number  is  diminished  by  9. 
What  is  the  number  ? 

Solution.     Let         x  represent  the  digit  in  the  units'  place ; 
then,  by  the  first  condition, 

5  —  a?  =  the  digit  in  the  tens'  place, 
and  10  (p  —  x) -\- X  =^  the  number,  [cf .  Ex.  13,  p.  61. 

and  10 X  +  (5  —  ic)  =  the  number  with  its  digits  interchanged. 

Hence,  by  the  second  condition, 

1 0  a;  4- T)  -  a;  =  1 0  (5  -  a?)  +  a;  -  9, 
whence  x  —  2,  and  5  —  a?  =  3. 

These  two  digits  are  found  to  satisfy  both  conditions  of  the 
problem ;  therefore  the  number  sought  is  32. 

EXERCISE  XXXI 

5.  John  has  14  cents  less  than  Henry ;  together  they  have 
60  cents.     How  much  money  has  each  ? 

6.  Divide  28  into  two  parts  whose  difference  is  4. 

7.  The  sum  of  two  numbers  is  63,  and  the  larger  exceeds  the 
smaller  by  17.     What  are  the  numbers  ? 

8.  If  16  is  added  to  a  certain  number,  the  result  is  the  same 
as  it  would  be  if  7  times  the  number  were  subtracted  from  56. 
What  is  the  number  ? 

9.  Of  four  given  numbers  each  exceeds  the  one  below  it  by 
3,  and  the  sum  of  these  numbers  is  58.     Find  the  numbers. 

10.  Divide  $2200  among  A,  B,  and  C  in  such  away  that  B 
shall  have  twice  as  much  as  A,  and  C  $  200  more  than  B. 

11.  I  take  a  trip  of  90  miles,  partly  by  train,  partly  by  trolley. 
If  I  go  42  miles  farther  by  train  than  by  trolley,  how  far  do  I 
go  by  each  ? 

12.  Three  boys  together  have  140  marbles.  If  the  second  has 
twice  as  many  as  the  first,  but  only  half  as  many  as  the  third, 
how  many  marbles  has  each  boy  ? 


60]  EQUATIONS  AND  PROBLEMS  65 

13.  After  taking  3  times  a  certain  number  from  11  times  that 
number,  and  then  adding  12  to  the  remainder,  the  result  is  less 
than  117  by  7  times  the  number.     What  is  the  number  ? 

14.  I  spend  $  2.50  for  3-cent  and  4-cent  stamps,  getting 25  more  of 
the  former  than  of  the  latter.    How  many  of  each  kind  do  I  buy  ? 

15.  A  man  who  is  32  years  old  has  a  son  who  is  8  years  old. 
How  many  years  hence  will  the  father  be  3  times  as  old  as  his 
son  (cf .  Ex.  9,  p.  61)  ? 

16.  The  sum  of  two  consecutive  integers  is  73.  What  are  the 
integers  (cf.  Ex.  14,  p.  61)  ? 

17.  Find  three  consecutive  integers  whose  sum  is  51.  Show 
that  the  sum  of  any  three  consecutive  integers  is  3  times  the 
second  of  these  integers. 

18.  The  difference  between  the  squares  of  two  consecutive 
integers  is  19.     Find  the  integers. 

19.  Find  two  consecutive  even  integers  whose  sum  is  98.  • 

20.  Find  the  even  integer  whose  square  subtracted  from  that 
of  the  next  higher  even  integer  leaves  52. 

21.  The  accompanying  diagram  represents 
the  floor  of  a  room.  If  the  perimeter  (the  dis- 
tance around  it)  is  5  times  the  width,  how  wide 
is  the  floor  ?  how  long  ?  How  many  square 
yards  in  its  area  ? 

22.  The  length  and  breadth  of  a  rectangular  floor  differ  by 
5  ft. ;  the  perimeter  is  60  ft.  Find  the  dimensions  and  area  of 
this  floor ;  also  make  an  accurate  diagram  of  the  floor. 

23.  If  each  side  of  a  square  lot  were  increased  by  2  yd.,  the 
area  of  the  lot  would  be  increased  by  96  sq.  yd.  Find  the  side 
of  the  given  lot.     Draw  an  appropriate  diagram. 

24.  A  certain  rectangle  is  5  ft.  longer  than  it  is  wide ;  if  each 
dimension  were  increased  by  2  ft.,  the  area  would  be  increased  by 
38  sq.  ft.  Find  the  length,  the  breadth,  and  the  area  of  this 
rectangle. 


(aH-6)feet 


66  HIGH  SCHOOL  ALGEBRA  [Ch.  V 

25.  Five  boys  agreed  to  purchase  a  pleasure  boat,  but  one  of 
them  withdrew,  and  it  was  then  found  that  each  of  the  remaining 
boys  had  to  pay  $  2  more  than  would  have  been  necessary  under 
the  original  plan.     How  much  did  the  boat  cost  ? 

26.  A  laborer  was  engaged  to  do  a  certain  piece  of  work  on 
condition  that  he  was  to  receive  $2  for  every  day  he  worked,  and 
to  forfeit  50  cents  for  every  day  he  was  idle ;  at  the  end  of  18 
days  he  received  $28.50.     How  many  days  did  he  work? 

27.  A  number  consists  of  two  digits  whose  sum  is  8;  and  if 
36  is  subtracted  from  this  number,  the  order  of  its  digits  is 
reversed.     What  is  the  number? 

28.  In  a  certain  two-digit  number  the  tens'  digit  is  twice  the 
units'  digit,  and  the  number  formed  by  interchanging  the  digits 
equals  the  given  number  diminished  by  18.    What  is  the  number  ? 

29.  A  two-digit  number  equals  7  times  the  sum  of  its  digits ; 
the  tens'  digit  exceeds  the  units'  digit  by  3.     Find  the  number. 

30.  What  principal  at  5  %  interest  yields  an  annual  income  of 
$250  ?  What  principal  at  4  %  simple  interest  amounts  in  5  years 
to  $2400  (cf.  Ex.  12,  p.  61)? 

31.  I  make  two  equal  investments,  one  at  6%,  one  at  4%.  If 
the  difference  in  the  annual  income  from  the  two  is  $80,  find  the 
total  sum  invested. 

32.  Two  trains  which  travel,  respectively,  30  and  50  miles  an 
hour,  start  toward  each  other  at  the  same  time  from  two  cities 
240  miles  apart.     How  long  before  they  meet  ? 

Suggestion.  Let  x  =  the  number  of  hours  before  they  meet ;  then 
30x  +  50x  =  240  (cf.  Ex.  15,  p.  61). 

33.  Two  bicyclists  ride  toward  each  other  from  towns  104  miles 
apart,  the  first  at  the  rate  of  12  miles  an  hour,  the  second  at  the 
rate  of  14  miles  an  hour.  If  they  start  at  the  same  time,  how 
long  before  they  meet  (cf.  Ex.  15,  p.  61)  ? 

34.  Two  bicyclists,  A  and  B,  whose  rates  are,  respectively,  12 
and  15  miles  an  hour,  start  from  the  same  town  and  rid^  in  the 
same  direction.  If  A  starts  1|  hours  before  B,  how  long  before 
B  overtakes  him  ? 


60]  EQUATIONS    AND    PROBLEMS  67 

35.  A  walks  m  miles  at  the  rate  of  3  miles  an  hour,  returning 
at  the  rate  of  2i  miles  an  hour.  If  the  entire  walk  is  made  in 
5^  hours,  what  is  the  value  of  m  ? 

Hint.     The  first  half  of  the  walk  can  be  made  in  —  hours,  the  second  in 

^  hours  (cf.  Ex.  16,  p.  61). 
5 

36.  A  tourist  climbs  a  certain  mountain  at  an  average  rate  of 
2  miles  an  hour,  and  descends  at  an  average  rate  of  3  miles  an 
hour.     If  the  round  trip  takes  6  hours,  how  long  is  the  path  ? 

37.  The  diiference  of  the  radii  of  two  circles  is  4  inches ;  the 
sum  of  their  circumferences  is  88  inches.  Find  the  radius  of 
each.  (The  circumference  of  a  circle  equals  27r  times  its  radius ; 
and  7r  =  3|,  approximately.) 

38.  Divide  $351  among  three  persons  in  such  a  way  that  for 
every  dime  the  first  receives,  the  second  shall  receive  25  cents, 
and  the  third  a  dollar. 

39.  Divide  48  into  two  parts  such  that  twice  the  larger  part 
equals  10  times  the  smaller  part  (cf.  Ex.  34,  p.  7). 

40.  Three  times  Harry's  age  equals  5  times  the  age  of  his 
sister ;  the  sum  of  their  ages  is  24  years.     How  old  is  each  ? 

41.  A,  working  alone,  can  do  a  certain  piece  of  work  in  3  days; 
B,  in  6  days.  In  how  many  days  can  they  complete  it,  working 
together  (cf.  Ex.  17,  p.  62)? 

Hint.     Let  x  =  the  required  number  of  days  ;  then  -  =  -+-. 

aj     3     o 

42.  Solve  Prob.  41,  if  A  can  do  the  work  in  8  days,  B  in  6  days  ; 
also,  if  A  can  do  the  work  in  4|^  days,  B  in  4  days. 

REVIEW  EXERCISE-CHAPTERS  J-V 

1.  Define :  negative  number,  absolute  value  of  a  number,  coef- 
ficient, exponent,  term,  polynomial,  degree  of  a  term,  finite  num- 
ber, equation,  root  of  an  equation,  identity.  Illustrate  each  of 
your  definitions; 

2.  Define  and  illustrate :  inverse  operations,  multiplication, 
division.  Point  out  at  least  one  advantage  which  the  definition 
of  multiplication  given  in  §  7  (ii)  has  over  that  in  §  7  (i). 


68  HIGH   SCHOOL  ALGEBRA  [Ch.  V 

3.  If  distances  above  sea  level  are  called  positive,  what  would 
—  25  feet  mean?  +35  feet?  What  is  the  difference  between 
these  elevations  ? 

4.  A  walks  east  at  the  rate  of  3  miles  an  hour,  B  at  the  rate 
of  —2  miles  an  hour.  How  long  before  the  two  are  15  miles 
apart?     Illustrate  by  a  drawing. 

5.  Translate  into  algebraic  language : 

(1)  The  number  formed  by  interchanging  the  digits  of  a  cer- 
tain two-place  number  exceeds  the  number  itself  by  18. 

(2)  A  certain  number  diminished  by  5  %  of  itself  equals  76. 

(3)  The  sum  of  two  consecutive  even  integers  equals  half  the 
difference  of  their  squares. 

6.  Point  out  at  least  one  advantage  in  using  letters  to  repre- 
sent numbers. 

7.  How  are  two  or  more  similar  monomials  added?  State  a 
rule  for  subtracting  one  polynomial  from  another. 

8.  How  may  a  parenthesis  which  incloses  several  terms,  and 
which  is  preceded  by  the  minus  sign,  be  removed  without  affect- 
ing the  value  of  the  expression  ?     Why  ? 

9.  State  the  law  of  signs  for  multiplication ;  for  division. 
What  powers  of  —3  between  the  1st  and  12th  are  negative  in 
sign  ?     Why  ?     Find  the  value  of  (  - 1)3 .  (  -  10)^  --  (  -  5)4. 

10.  State  the  exponent  law  for  multiplication ;  for  division. 
By  reference  to  these  laws  find  the  value  of  ^^"^'^' '  ^^"'^^^". 

If  c  =  9,  d  =  -5,  e  =  -2,/=-2,^  =  l^  find  the  value  of; 

11.  5c  +  3d-e4-/-c--/+2c.3^--e  +  8/. 

12.  4 c'f  -i-6eg^-\-d^-\-5[e- 3/+  c'-3 ed]. 

13.  -^i^llc-(12g'-6efg)^-\-(c'-^d^^{c  +  dy. 
Perform  the  following  indicated  operations : 

14.  (x—5y){4:y  —  x).  17.    (a^  — 1)1 

15.  (am  — en) (3  a -f  5  c).  18.    (m^  ~mn-{-7')(m^-{-mn  —  r). 

16.  (k-2qf.  19.    (62-  +  c^)(&2-_c-). 


50]  REVIEW  EXERCISE  69 

20.  (a^«  +  2/^-i)(a:-l).  22.    (x"^+* -5x''+'+6)--r(x'^+^-3). 

21.  (r2"'  —  r"*H-l)  (3 +  ?•'»-').        23.    {x^' -y^'^)  ^  (y^ —  x"). 

24.  (1x2+ ia;-i)  +  (f  a.'- 3  x^-r"')- (-2x^  +  1^-1).' 

25.  4p  -  [p2  4. 22)r  -  (2  q  +p')  j  +  (1  -j^r  +  g). 

26.  5m  —  Sn—\  —  7n-\-m  —  5n  —  3ml. 


27.  2a;-a;-2/  +  22-[-S-(y  +  42;)|]. 

28.  a(6  +  c  —  cZ)  —  6(a  —  2  c  +  d)  —  3  c(—  a  —  d). 

If  itf=7a6-3Z>2_4a2,  ^=36^-4  a^- «?>,  P=a-b,  and 
Q  =  ?,*  _  4  a^ft  -  4  a6^  -h  6  a262  +  a^  find  the  value  of : 

29.  M-\-N.  33.    J»f2.  37.    Qh-P. 

30.  Jf-iV:  34.    itfP.  38.    Q^M. 

31.  |(4jlf+iV^).        35.    MN.  39.    2Q-NP. 

32.  P^-5N.  36.    ^-J-iJf.  40.    Q--P'^  +  3P2. 

41.  Check  your  work  in  Exs.  32-37.  What  two  checks  may 
be  used  for  Ex.  30  ?  for  Ex.  36  ? 

42.  In  Ex.  44  below,  insert  the  second  and  third  terms  in  a 
parenthesis  preceded  by  +  ;  place  the  fourth  and  fifth  terms 
under  a  vinculum  preceded  by  —  (see  footnote,  p.  37). 

43.  In  Ex.  45  below,  inclose  the  first  three  terms  in  a  paren- 
thesis preceded  by  —  ;  place  the  last  two  under  a  vinculum  pre- 
ceded by  + . 

In  each  of  Exs.  44-49,  collect  the  coefficients  of  r,  s,  and  t 
(cf.  Exs.  24-26,  p.  47). 

44.  2  a^s  —  3  ar  —  6s  —  cs  —  4  r. 

45.  -r-\-5fh  +  2s+fh-lft. 

46.  -nH-{-5<^ds-2hH-'ieft-\0<?cls, 

47.  t  -f  aV  +  6=^s  H-  r  -i-  s  —  2  c?  +  2  hs  -{-cH  —  2  ar. 

48.  (a  H-  2  c)s  -f  (c  +  d)r  +  2  cs  —  5dr  —  7  ds  —  er. 

49.  (m^  -  l)s  -  2(1  -  ny  +  nV  -3s-  (m*  -  2  mhi^)r  +  91^^. 

50.  Multiply  x^ -\-  {a -{- c)x  hj  X  —  c -,  hj  x^ -j- ex -\-  k. 


70  HIGH  SCHOOL  ALGEBRA  [Ch.  V 

51.  Divide  2  a^  -f-  2  wV  —  n^x  —  mn^  —  a^(2  m  -f-  2  n)  —  wa^  — 
2  mnic  +  mna;^  by  —x^  +  {m-{-  n)x  +  mn  (cf .  Ex.  47,  p.  51).  Check 
your  result  by  multiplication. 

Solve,  and  check  the  roots: 

52.  x{2  x-B)-51  =  2  x{x-\-l).  54.  |a;-(|a;  +  i)  =  -a;  +  f 

53.  ^±2  =  ?^±3^_2^.       55.    |(.-2)(.-l)=^^-^^ 

5  3  2  4 

56.  Four  times  the  number  of  seniors  in  a  certain  school 
exceeds  the  number  of  freshmen  by  15.  If  the  total  enrollment 
in  the  two  classes  is  130,  find  the  size  of  each  class. 

57.  A  certain  number  is  subtracted  from  50  and  42  in  turn.  If 
\  of  the  first  remainder  equals  ^  of  the  second,  find  the  number. 

58.  If  the  population  of  a  town  has  increased  30  %  in  the  last 
10  years,  and  is  now  5200,  find  the  population  10  years  ago. 

59.  A  man  spends  \  of  his  income  for  living  expenses  and 
insurance,  ^^  for  books,  -^^  for  travel,  -^^  for  charities.  If  he 
saves  $425,  what  is  his  income  ? 

60.  A  certain  rectangle  is  8  ft.  longer  and  5  ft.  narrower  than 
a  given  square,  and  its  area  exceeds  that  of  the  square  by  5  sq.  ft. 
Find  the  side  of  the  square.     Draw  an  appropriate  figure. 

61.  A  and  B  together  can  do  a  certain  piece  of  work  in  6  days. 
If  A  can  do  it  alone  in  10  days,  in  how  many  days  can  B  do  it  ? 

62.  Silk  marked  to  sell  at  a  gain  of  33|^  %  has  its  marked  price 
reduced  20  %,  and  then  sells  for  80  cents  a  yard.     Find  its  cost. 


CHAPTER   VI 

TYPE    FORMS   IN   MULTIPLICATION  —  FACTORING 
I.     SOME   TYPE  FORMS   IN  MULTIPLICATION 

51.  Type  forms.  Although  all  exercises  in  multiplication 
and  division  of  integral  polynomials  can  be  readily  solved 
by  §  38  and  §  39,  yet  there  are  a  few  special  cases  of  these 
operations  which  occur  so  frequently  in  practice  that  it  is 
well  worth  one's  while  to  memorize  them ;  they  are  often 
spoken  of  as  type  forms.  Some  of  these  type  forms  are  con- 
sidered in  the  next  few  paragraphs. 

52.  Square  of  a  binomial.  Let  a  and  h  represent  any  two 
numbers  whatever;  then  by  actual  multiplication  (§  33), 

(«  +  ^»)  (a  +  6)  =  ^2  4-  2  a6  +  ^2, 
and  (a  -  6)  (a  -  6)  =  a2  -  2  a6  +  52 . 

i.e.,  (a+6)2  =  a2  +  2a6  +  62,  I 

and  (a_6)2  =  a2-2a6  +  62,  II 

whatever  the  numbers  represented  by  a  and  h. 

Translated  into  common  language,  I  becomes : 

The  square  of  the  sum  of  any  two  numbers  equals  the  square 
of  the  first  number^  plus  twice  the  product  of  the  two  numbers^ 
plus  the  square  of  the  second  number. 

The  student  may  translate  II  into  common  language. 

By  means  of  I  and  II,  we  can  now  write  down  (without 
actually  performing  the  multiplication)  the  expanded  form 
of  the  square  of  any  binomial  whatever.     Thus  : 

Ex.  1.  (m  +  3)2  =  m^  +  6  m  +  9. 

Ex.  2.  (x  —  yy  =  x^~2xy-\-  ?/. 

Ex.  3.  (2  s  -  3  ty  =  (2  sy  -  2(2  s)(3  t)  +  (3  ty  =  4  s2  -  12  st  +  9  tK 

71 


72  HIGH   SCHOOL  ALGEBRA  [Ch.  VI 

EXERCISE  XXXII 

Expand  the  following  (check  your  work  as  teacher  directs)  : 

4.  {x^yf.  12.  (1+2^)'.  20.  (la -2)2. 

5.  {m  +  n)\  13.  {l-vf,  21.  {\yJ^Qf. 

6.  Qi-^kf.  14.  (2  a; -6)2.  22.  (5a^-f)2. 

7.  (w  +  it;)l  15.  (a.  +  7ic)2.  23.  (5m  +  |w)2. 

8.  (a-p)l  16.  (3m^-2)2.  24.  (2  a^x  +  3  62/^)2. 

9.  (c-/i)l  17.  (2g^-5hy.  25.  (5  rs  -  3  r^s^)^ 

10.  (x  +  Sy.  18.    (11-7A;)2.  26.    (ic'»  +  r)2. 

11.  (a  -  5)2.  19.    (4  b^  -h  1)2.  27.    (3  a"  -  2  s^f. 

28.  Expand:    (a -\- b  -  5)%    i.e.,    ](a -^  b) -5\^',    (c  +  2H-d)2; 
(_2c-(Z  +  e«)2;  (7-ha2-c)2;  (3a.-«-p-5)2. 

29.  Since  a  —  b  =  a^(  —  b),   show  that  II,    §  52,  is  included 
under  I. 

30.  What  must  be  added  to  x^-^6x  to  make  it  the  square  of  a;+3 ? 

31.  What  must  be  added  to  a'*  +  a262  -f-  b*  to  make  it  the  square 
oia'-i-b'? 

32.  What  must  be  added  to  25  — 10  a^  to  make  it  the  square  of 

33.  What  must  be  added  to  ic^  +  2  x*y^  +  4  /  to  make  it   the 
square  of  a;'' +  2/? 

By  the  method  of  §  52  write  down  the  squares  of  the  following 
numbers  : 

34.  16,  i.e.,  10  +  6.  36.   28.  38.    71. 

35.  19,  i.e.,  20  - 1.  37.   43.  39.   83. 

40.    Expand  (a  +  1)2 ;  also  (—a  —  1)2.     Compare  and  explain. 

53.   Product  of  sum  and  difference.     If  a  and  b  represent 
any  two  numbers  whatever,  then,  by  actual  multiplication, 

whatever  the  numbers  represented  by  a  and  h. 

The  student  may  translate  this  formula  into  common  lan- 
guage (cf.  §  52). 


52-54]  TYPE  FORMS   IN  MULTIPLWAriON  73 

EXERCISE  XXXIII 

Write  the  following  products  by  inspection  and  check  results : 

1.  ix^y){x-y).  12.    (4+a«)(4_,,3), 

2.  (7n-\-n){m  —  7i).  13.    (2/"»  — 11)(^'" +  11). 

3.  {^x  +  y){^x-y).  14.    {ax''  +  W){ax''-W). 

4.  {x-2y)(x  +  2y).  15.    \{x-y)  +  z\\{x-y)-z\. 

5.  (4a  +  15Z/)(4a-15?>).  16.    K«  + &)  + cn(«  + &) -cj. 

6.  (G  /)  —  5  q)  (6  ji)  +  5  q).    *  17.    (m  +  n  +p)(m  +  n  — p). 

7.  (2a;2/— 7)(2a;?/  +  7).  18.   (c  -  d  +  5)(c  — d  — 5). 

8.  (4m2-3n)(4m2  +  3w).  19.    j2-(a;4-2/)n2 +(^4-2/)S. 

9.  (9  +  5pV)(9  — 5pV).  20.    (7  +  m  +  rz)(7  — m  — w). 

10.  (a^  +  i/)(a^-i/).  21.    {a-h^x){a-\-h-x). 

11.  (10mn-6)(10mn  +  6).  22.   (2  A:  -  Z  +  3)(2  A:  +  Z-3). 

23.  (9lx''-4.f)-^(^x-2y)  =  ?     Why? 

24.  (16a2-2562)-f-(4a  +  56)  =  ?     Why? 

25.  (a;^-2/')^(a;^-2/0  =  ?     Why? 

26.    (a;«-2/*)-(ar^-/)=?  27.    (a;i«  -  ^/S) -^  (a^^  +  2/')  =  ^ 

28.   Find,  by  the  above  method,  the  product  of  22  by  18,  i.e., 
of  20 +  2  by  20-2;  of  17  by  23;  of  42  by  38;  of  56  by  44. 

54.   Product  of  binomials  having  a  common  term.     By  ac- 
tual multiplication, 

(2^+3)(:r+5)  =  a;2  +  8:r4-15  =  rr2^(3  +  5)2:4-15; 
and    (a:+3)(a:-5)  =  a:2-2a;-15  =  a;2  +  (3-5>-15. 
And,  in  general, 

(jr  +  a)(jr  +  6)=  jr2  +  (a  +  6)jr  +  ab, 

whatever  the  numbers  represented  by  a,  h,  and  x. 
Translating  this  formula  into  words,  it  becomes: 
The  product  of  two  binomials  having  a  term  in  common  equals 

the  square  of  the  common  term,  plus  the  algebraic  sum  of  the 

unlike  terms  multiplied  by  the  cornmon  term,  plus  the  product 

of  the  unlike  terms. 


74  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

EXERCISE  XXXIV 
Write  down  the  following  products  (check  as  teacher  directs) : 

1.  (a  +  5)(a  +  7).  16.    {xy  -  ^)(xy +  U). 

2.  (a-5)(a-7).  17.    (-8-fmV)(2  +  mV). 

3.  (a  +  5)(a-7).  18.    (s4- 7'2)(3s  + r^). 

4.  (a_5)(a  +  7).  19.    \{l +  m) -2\\{l^m) -^, 

5.  (2/-c)(2/  +  2c).  20.    5(Z  +  m)  +  8n(^  +  m)-15i. 

6.  (a^-f.4)(ar'  +  5).  21.    (m -n- 5)(m-n- 9). 

7.  (a^  +  4)(^2_5>)^  22.    (s-«  +  4)(s-^-4). 

8.  (a^_4)(aj2_5)^  23^    (,.p_  10)(r^  + 15). 

9.  (a^_4)(aj2-|.5).  24.    (a;^- +  3)(a^2- -7). 

10.  (3  4-m)(5  +  m).  25.    (3  a^-a;)(2.T  +  3  a^). 

11.  (6  +  a)(c  +  a).  26.    (&^  +  2/)( -c  +  2/). 

12.  (2a;  +  l)(-5  +  2aj).  27.    \^{axY  +  2\\^{a.xf  -\-l\. 

13.  (a-&)(a-c).  28.    (5  -  4  i«y )  (5  +  a^y ) . 

14.  (4  +  3a)(-6+3a).  29.  .(m2-c)(- 7  m^- c). 

15.  (4s2-5)(4s2-|-i).  30.    (3j9-g-7)(3p-g  +  7). 
31.  When  6  =  a,  what  does  the  formula  of  §  54  become  ?    What 

does  it  become  when  b  =—  a?     Are  the  formulas  of  §§52  and 
53  only  special  cases  of  (x -{-  a)  (x -\- b)  =  a^  +  (a -\-  b)x  +  a&? 

55.  Product  of  two  binomials  whose  corresponding  terms  are 
similar.     By  actual  multiplication  we  obtain 

^x  -1y 
15  x^  +  20  xy 

—    ^  xy  —%y^ 
15x^-{-Uxy-Sy^ 

Here  the  term  14  xy  is  the  algebraic  sum  of  the  "  cross 
products  "  5x'4:y  and  —  2  y  •  S  x. 

With  a  little  practice  the  final  product  of  two  such  bi- 
nomials may  be  written  down  by  inspection,  i.e.^  without 
first  writing  the  partial  products. 


54-66]  TYPE  FORMS  IN  MULTIPLICATION  76 

EXERCISE  XXXV 
Write  the  following  products  by  inspection : 

1.  (3a;  +  2)(4a;-3).  4.    (o- 11)(3  a- 1). 

2.  (5m-l)(2m-3).  5.    (3x-^2 y){Ax-\-3y), 

3.  (2r  +  5)(r-5).  6.   (x-S  y)(5  x-h  6y). 

7.  In  each  of  the  above  products,  how  is  the  first  term  ob- 
tained ?   the  third  ?   the  second  ? 

8.  What  is  meant  by  the  expression  "  cross  products  "  as  used 
in  §  55  ?     Illustrate  from  Ex.  3,  above. 

Write  down  the  following  products  by  inspection,  and  check 
results  as  the  teacher  directs  : 

9.  (7-2m)(7-m).  18.    (ia-2)(|a+4). 

10.  (3  -  4  a)  (4 -f  3  a).  19.  {x-{- a){x-^b). 

11.  (9x  —  2y)(x-\-y).  20.  (3x+ c){x-^d). 

12.  (2a-4  62)(5a-6  62).  21.  (3x-c)(x-d). 

13.  (7(y'-{-d'){3c^-\-Sd^.  22.  (3  x-{- c){5  x  +  d). 

14.  (a"'-2e)(a'"-e).  23.  (3  x-c){5  x-d). 

15.  (6ic^-f4)(3a;^-2).  24.  (ky -\-l)(ny-l). 

16.  (11-7  cd^){6-{- 3  cd^).  25.  (Jcy -^a){ny +  b). 

17.  (ic  +  2)(ic  +  l).  26.  Qcy-a){l-cy). 

56.  The  square  of  any  polynomial.     By  actual  multiplica- 
tion it  is  found  that 

(a  4.  5  +  c)2  =  ^2  4-  52  +  c2  +  2  a6  +  2  a^  +  2  he, 
(^a-^b-{-c-\-dy  =  a'^-{-P  +  c'^-{-d^  +  2ah-\-2ac  +  2ad 
+  2bc-{-2bd-h2cd, 
(^a-]-b-^c+d+ey=a^-^h^-{-(P  +  d^-{-e^  +  2ab-^2ac-{-2ad 
+  2ae-{-2bc-{-2bd+2be-\-2cd-\-2ce-{-2de, 

and  so  on  for  any  polynomials  whatever;  that  is,  The  square 
of  any  'polynomial  whatever  equals  the  sum  of  the  squares  of 
all  the  terms  of  the  poly)fiomial,  plus  twice  the  product  of  each 
term  by  all  the  terms  that  follow  it  (for  proof  see  §  209). 

HIGH  SCH.   ALG.  — 6 


76  HIGH  SCHOOL   ALGEBRA  [Ch.  VI 

EXERCISE  XXXVI 

Expand  by  inspection  (check  as  teacher  directs)  : 


1. 

{c  +  d  +  ey. 

12. 

(4a3-&_5)2. 

2. 

(m  +  n  —  sf. 

13. 

(5-^x-fy, 

3. 

(a-b-cf. 

14. 

(-5-x  +  yy. 

4. 

(m  +  r  +  l)2. 

15. 

(a  —  b-\-c  —  dy. 

5. 

(^m-r-Sy. 

16. 

(ax-\-by-\-czy. 

6. 

(2x-\-y-^zy. 

17. 

(ia-ic  +  iey. 

7. 

(2x-j-3y-zy, 

18. 

(mn  —  np—pqy. 

8. 

{2x-3y-\-zy. 

19. 

(abx  —  acy  —  bczy. 

9. 

l-(2x  +  3y^l)y. 

20. 

(2x-3y  +  4.z-ay. 

10. 

(3c2  +  d-4)l 

21. 

(x^-\-x  +  iy. 

11. 

(4.a'-\-b'  +  Sc'y. 

22. 

(/+m+w+p+^4-r+s)2. 

23.  Could  any  of  the  above  products  have  been  found  by 
means  of  formulas  alrfea^dy  used  (cf.  §  52,  also  Ex.  28,  p.  72)  ? 

24.  Give  a  rule  for  writing  down  the  square  of  any  polynomial 
whatever.  What  does  this  rule  become  when  the  polynomial  is 
a  binomial  (cf .  §  52)  ? 

57.  Cube  of  a  binomial.  The  cube  of  a  binomial  is  another 
product  which,  because  of  its  frequent  occurrence,  should  be 
memorized.     By  actual  multiplication  we  obtain 

and  (a-by  =  a^-3aH-{-Zab^-b^ 

whatever  the  numbers  represented  by  a  and  b. 

By  means  of  these  formulas  (which  the  pupil  should  trans- 
late into  words)  we  may  write  by  inspection  the  cube  of 
any  binomial  whatever. 

Note.  §  52  and  §  57  are  particular  cases  of  what  is  known  as  the 
binomial  theorem;  this  theorem  is  considered  in  §  112. 

Ex.1.    (x-\-2y  =  a^-\-3  x''2-\-3x-2^-\-2^=x'-{-6x^-{-12x-\-S. 
Ex.  2.    (2  a-5  by  =  (2  a)«-  3  (2  ay  •  (5  b)  +3  (2  a)  •  (5  by-  (5  by 
=  8  a«-  60  a'b  + 150  ab'  - 125  b'. 


EXERCISE  XXXVII 

Expand  the  following 

:  expressions : 

3.   (x-^yf. 

7.    (c  +  l)«. 

11. 

4.   (m-ty. 

8.    (a -3)3. 

12. 

5.   (2x-yy. 

9.    (d'  +  c^y. 

13. 

6.    (z-Syy. 

10.    (2yz-5y. 

14. 

56-68]  FACTORING  77 


(l  +  2m)3. 
(3a2-2  67^ 
(_5_v)3. 

15.  What  is  the  difference  in  meaning  between  "the  cube 
of  the  sum  of  two  numbers"  and  "the  sum  of  the  cubes  of  two 
numbers  "  ?     Illustrate,  using  3  and  4  as  the  two  numbers. 

16.  Give  a  rule  for  finding  the  cube  of  the  difference  of  two 
numbers. 

17.  Expand  (c-\-dy,  also  (—c—dy.  If  we  change  the  signs  of 
an  expression,  do  we  change  the  signs  in  its  cube  ?     Why  ? 

II.    FACTORING 

58.  Definitions.  In  a  broad  sense,  any  two  or  more  num- 
bers whose  product  is  a  given  number  are  factors  of  that 
number.  Thus,  since  J -1^.  15=6,  therefore  ^,  |^,  and  15 
are  factors  of  6;  so  also  are  ^,  18,  and  ^;  the  important 
factors  of  6  are,  however,  2  and  3. 

In  order  to  exclude  fractional  and  other  unimportant  fac- 
tors, we  shall  (as  it  is  customary  to  do)  define  factors  thus: 

The  factors  of  a  number  or  algebraic  expression  are  its 
rational*  integral  exact  divisors. 

JEJ.g.^  the  factors  of  3  a;  (a^  —  b^}  are  S,  x,  a-\-  6,  and  a  —  b, 
as  well  as  the  product  of  any  two  or  more  of  these. 

Observe,  too  that  if  3,  a  +  6,  etc.,  are  factors  of  any  given 
expression,  then  —  3,  —  (a-f-J),  etc.,  also  are  factors  of  this 
expression. 

A  factor  (or  expression)  is  said  to  be  prime  if  it  contains 
no  factors  except  itself  and  1 ;    otherwise,  it  is  composite. 

*  An  expression  \\raUonal  with  regard  to  a  particular  letter  if  it  contains 
no  indicated  root  of  that  letter  (see  §  113). 


78  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

By  factoring  a  number  (or  expression)  is  usually  meant 
the  procetis  of  separating  it  into  its  prime  factors. 

Factoring  an  expression,  as  will  appear  later,  often  greatly 
simplifies  algebraic  work ;  it  is  therefore  important  that  the 
pupil  should  early  master  those  cases  of  factoring  which 
present  themselves  most  frequently. 

59.  Factors  of  a  monomial.  The  literal  factors  of  a  mono- 
mial are  evident  by  inspection,  and  the  factors  of  the  numer- 
ical coefficient  are  found  as  in  arithmetic. 

E.g.y  the  factors  of  30  a^x^y^  are  2,  3,  5,  a,  oi?,  and  t/^.  (The  ic-and 
^/-factors  are  as  evident  in  the  forms  ni?  and  y^  as  from  x-x-  x  and 

yy-) 

60.  Monomial  and  polynomial  factors  of  a  polynomial.    If  a 

polynomial  contains  a  monomial  factor,  the  latter  is  readily 
discovered  by  mere  inspection. 

E.g.,  in  12aVH-4a6a^?/  — 8aicy,  it  is  seen  that  each  term  contains 
the  factor  4  ax^,  hence  (see  §  38)  the  other  factor  is  Sax-\-by—2y^; 
i.  e.,  12  a  V + 4  abx^y  —  8  ax^y^ = 4  aar^  (3  ax  -{-by— 2  y^). 

To  factor  a  polynomial  completely/  requires  (1)  the  removal 
of  all  monomial  factors,  and  (2)  the  factoring  of  the  poly- 
nomial thus  freed  from  its  monomial  factors.  The  simpler 
cases  of  (2)  are  considered  in  the  next  few  articles;  (1) 
may  always  be  accomplished  as  above. 

EXERCISE  XXXVIII 

Factor : 

1.  Qa'x^.  3.  42  s¥.  5.  408  mVt/l 

2.  WmjA^  4.  210  2/V.  6.  572  a'(fwv\ 

7.  The  expression  5  a  —  10  6  +  30  a^  has  what  monomial  fac- 
tor? what  polynomial  factor?  How  do  you  find  the  former? 
the  latter  ? 

Separate  the  following  expressions  into  their  monomial  and 
polynomial  factors,  and  check  your  results : 


58-61]  FACTORING  79 

8.  17a;2-51^.  14.  3  a' -  6  a'b  +  a'b\ 

9.  4  ar'  —  6  x-y.  15.  7nIn'^-\-  m^w^  +  m^/i^. 

10.  4a-^62-26a-6l  16.  3  r^  -  12  r^s^  +  6  rs^ 

11.  10  mhi^  —  15  m%\  17.   ac  —  bc  —  cd  —  abed. 

•    12.    -16x--2abx.  18.  32  .^ry - 28  a^y  +  12 .ti/. 

13.   15a;^-10ar^  +  25a^.  19.   Uxyz^-2  afyh''-{-Sxyh\ 

20.  60  mhih^  —  45  m^nh'^  -\-  90  m*n^r^. 

21.  12a;267/-18a.7y«6  +  24a;y?>*. 

22.  14  a^mii^  —  21  aSnhi^  —  49  a*mn^. 

23.  35  c^cZar^  +  ^  c\?  V  -  55  c^d  V. 

24.  51i«?/V-68ar^/22_^85a;y^l 

25.  52  a%V- 65  ^6=^6^  + 91  a2?>V. 

26.  Write  (m  +  7?.)^  —  3(m  +  7i)^  +  (m  +  n)  as  the  product  of  two 
factors,  one  of  which  is  m  +  71. 

27.  Write  2(3  0- -  1)2  -  5(3  a; -1)  + 4(3  0^-1)3  as  the  product 
of  two  factors;  also  6  (2 -a)^  -  8  (2  -  a)^  -  12  (2  -  a)«;  also 
x\a  -  c)-(l-3x')  (a-c)-(a-  c). 

28.  If  —  5  mhi^  is  one  factor  of  10  m'^71^  —  15  mhi^,  what  is  the 
other  (cf.  Ex.  11)  ?  Factor  again  the  expressions  in  Exs.  12-16, 
in  each  case  taking  the  monomial  factor  as  negative. 

61.  Factoring  by  means  of  type  forms.  Expressions  of  the 
type  a^  +  2  a6  +  6^.  Factoring  being  the  inverse  of  multipli- 
cation, it  follows  that  to  every  case  of  multiplication  there 
corresponds  a  case  of  factoring.  Ease  in  factoring,  as  in 
every  inverse  process,  depends  upon  a  ready  knowledge  of 
the  corresponding  direct  process. 

Thus,  if  we  promptly  recognize  the  form 

a2  +  2  a5  +  52,  [see  §  52 

then  we  can  as  promptly  write  down  its  factors,  viz. : 
a-\-h  and  a-{-h. 
So,  too,  the  factors  of 

^2  _  2  a5  +  52 
are  a  —  h  and  a  —  h.  [see  §  52 


80  HIGH   SCHOOL  ALGEBRA  [Ch.  VI 

The  expressions  6  mn  +  m^  -f  9  w^  and  4  a;^  +  25  —  20  rr  be- 
long to  this  type  form,  for,  in  each  case,  two  terms  of  the 
trinomial  are  the  squares  of  certain  numbers,  and  the  third 
term  is  twice  the  product  of  these  numbers.  These  expres- 
sions may,  therefore,  be  written  as  (m  + 3  9i)(m  +  3  w)  and 
(2  2:  —  6)  (2  a?  —  5),  or  as  (m  +  3  ^)2  and  (2x  —  5)^,  respectively. 

EXERCISE  XXXIX 

Factor  the  following  expressions : 

1.  x'-'Zhx-^W.  8.   a'V'-2ah-\-l. 

2.  u^  +  2uw  +  'u^.  9.    l-12?/  +  36i/2. 

3.  a^-6a;  +  9.  10.    aJ^-4a^  +  4. 

4.  2/'-42/H-4.  11.   30a^  +  225  +  a;i».      * 

5.  l-\-2a  +  o?.  12.   9  a;2  - 12  a;2/2  +  4  2/V. 

6.  m^- 10  m +  25.  13.   6  a5cd  +  9  c^d^  _^  a^ftl 

7.  49  -  14  s  4- s'.  14.   4-36a^&2^81a^6l 

15.  What  first  suggests  to  you  that  a;^  +  9  ?/^  +  6  xy  may  be  the 
square  of  a  binomial  ?  How  do  you  test  the  correctness  of  this 
supposition  ?     When  is  a  trinomial  the  square  of  a  binomial  ? 

16.  Write  out  a  carefully  worded  rule  for  factoring  expressions 
of  the  types  ar  +  2ab-\-W  and  a^-2ab  +  h'^'?  How  do  we 
find  the  terms  of  the  binomial  ?  How  do  we  determine  the 
sign  by  which  they  are  to  be  connected  ? 

17.  Is  a*  +  2  c^Jf  —  W  the  square  of  a  binomial  ?     Explain. 
Factor  the  following  expressions,  and  check  your  work. 

18.  5  a7?  —  80  aa?  +  320  a.      [Remove  monomial  factor  first.] 

19.  7  n^  +  l^ahri'^l  o?h''n.  25.    a^P-6cM"  +  9  6^ 

20.  18a3t/-60a%  +  50a6Y        26.   ic^-ic  +  f 

21.  27  cV- 36  0^^^7134-12  c^c^V.     27.   l  +  f^s^  +  fs. 

22.  —  a^  +  2  0^2/  —  2/^.  28.   m2p+2  +  2  mP+V+3  +  7i2'+« . 

23.  —  m*  H-  2  m^n^  —  m^n^  29.   (a  -f  x)^  +  2(a  +  cc)  + 1. 

24.  a:2n4.4ajy4.42/6.  30.   i6_8(ic  +  2/)  + (»  +  2//. 

31.   9(m  — n)2  — 6a7(m  — n)  +  a;^ 


1. 

/-^^. 

2. 

y'-^z\ 

3. 

4  2/2  _  49  62. 

4. 

2^a?W-lQ>. 

5. 

9 /-I. 

6. 

225  a;^- 9/. 

(Jl-62]  FACTORING  81 

62.  Expressions  of  the  type  a^  —  6^.  From  §  53  it  follows 
that  the  factors  of  aP'  —  b^  are  a -\-  b  and  a  —b. 

Again,  the  expression  25  n^  —  9  t^  is  of  the  above  type, 
and  its  factors  are  5^4-3^  and  5  w  —  3  ^. 


EXERCISE  XL 

Factor  the  following  expressions  : 

7.    a^x-h^x.  13.  49-36a^/. 

a   36aV-81dl  14.  m^^-n^-. 

9.   i»2n_4  3^5  49a^/-16 

10.  121a^-36  6^  16.  64a^/-81. 

11.  64  0^2/'" -144  21  17.  2S^o^z^-f^z. 

12.  {x  +  yf-\.  18.  4.d?-{x-yy. 

19,  In  factoring  the  difference  of  two  squares  {e.g.,  a^—b^), 
how  are  the  terms  of  each  factor  found  ?  How  are  these  terms 
connected  in  the  first  factor  ?  in  the  second  ? 

20.  Write  a  rule  for  factoring  the  difference  of  two  squares. 

By  rearranging  and  grouping  terms  factor  the  following  ex- 
pressions, and  check  your  work : 

21.  b^-2bc-d^-{-c\                     28.  -  a^+ 6*-2  a^-l. 
[i.e.,62_26c+c2-d!2,i.e.,(6-c)2-d2].   29.  -  18  A:  +  81  +  A;^ -  25  ^^^^ 

22.  c^  +  2cd-{-d'-e'.  30.  -9  ^^  +  49-12  wv -4^^^ 

23.  x^-b'-2xy-^y\  31.  S cH^ - 4: -{- c* -\- 16  d\ 

24.  x^  +  4:xy-4:z'  +  4.yK^^^      32.  s^ -4.7^ -^f-2  st. 

25.  m2-6m  +  9-p2.     /  33.  -524- ^2_4^_4  ^^^ 

26.  l-s'-2st-t\  34.   36c2-aV-36  +  12aa;. 

27.  25-m24-2m7i-n2.  35.    - 22 a;?/ + 121  - 2^  +  aj^^/^. 
36.    Supply   the   required   factor    in   each    of    the    following: 

a'-b'  =  (-a-{-b)'(?)',    16x'-9y'  =  (-4rX-Sy).(  ?  ). 

May  the  factors  in  Ex.  4  be  written  (  — 5a6— 4)(— 5a6H-4)? 
Explain  (cf.  §§  18,  58). 


82  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

63.  Expressions  of  the  type  x^+  mx-\-  n.     From  §  54  it  fol- 
lows that  the  factors  oi  aP' -{-  (^a -\- b)  x -\- ab  are 
x-\-  a  and  x  -\-  b. 
Again,  since  the  expression  A;^  +  7  A;  +  12  may  be  written 
in  the  form  A;^  +  (3  +  4)A:  +  3-4,  therefore  its  factors  are 
A:  +  3  and  k  +  4:. 
So,  too,        j92  4-2jt?-15=jt?2+(5-3)jt?  +  5.(-3) 

=  (^  +  5)(p-3). 
From  these  illustrations  we  see  that  we  can  separate  the 
trinomial  x^  +  mx  -\-  n  into  two  binomial  factors  whenever 
we  can  separate  n  into  two  factors  whose  sum  is  m.  Hence, 
whenever  such  an  expression  as  x'^  +  mx  +  n  can  be  factored, 
its  factors  may  be  found  by  a  few  trials  —  the  number  of 
trials  never  exceeding  the  number  of  pairs  of  factors  of  n. 

EXERCISE  XLI 

I.  If  the  expression  cc^  +  5  aj  —  36  is  the  product  of  two  bino- 
mial factors,  what  is  the  product  of  the  unlike  terms  in  these 
two  binomials  ?  Have  these  terms  like  or  unlike  signs  ?  Why? 
What  is  the  sum  of  these  unlike  terms  ?  Is  the  larger  of  them 
positive  or  negative  ?     Why  ? 

Factor  the  following  expressions  and  check  the  work : 

ax  —  90  al 

nx^^i2x 

[i.e.,  x(x2-  17  a; +  72)]. 
2x'-Qx  +  4.. 
Uv^-S2v'\ 

8.  r- 122? +  35.  18-    ax'-i-Ta^x  +  ea^ 

9.  m2  +  15m  +  50.  19-   66  +  39/  +  3/. 
10.   Jc'-Sk-AO.                             20.    -a2-27  +  12a. 

II.  'y2-7v-18.  21.   a2&2_7a&  +  10 
12.    f  +  13  ^  -  30.               •  ^    U-e-,  (aby  -  l{ah)  +  10]. 


2. 

x--^x  +  2. 

13. 

0? 

3. 

x'-\-x-Q. 

14. 

v' 

4. 

:^-x-2. 

15. 

:^ 

5. 

s2+_i2s4-36. 

\i. 

6. 

f^^y^h. 

16. 

2i 

7. 

o2  +  7a-30. 

17. 

v' 

63-64]  FACTORING  83 

22.  4ic2  +  4a;-3  30.    ^-'"-24r  +  63. 
[i.e.,  (2a:)2  +  2(2  x)  -  3].  31.    ^^2  _  ^^  ^^^  _^  28  n\ 

23.  4a;2-8a;-21.  32.  s'^-st-4t2f. 

24.  Oo.'^  +  eaj-S.  33.  -12a;2/2  +  ar/  +  32  22. 

25.  9x2-21a;-8.  34.  (m  +  ri)^^- 7(m  +  n)  +  6. 

26.  9x'  +  21x  +  U.  35.  (s-A;)2-26(s-A:)  +  69. 

27.  16ar^-56ic  +  33.  36.  Q-y-y\ 

28.  15  +  32a;H-16a;2^  37.  r^  -  {b  -  f)r  -  bf. 

29.  25a:2_8_;l^()^  33^  ic2  + (3a- 2  5)a;-6a&. 

64.   Expressions  of  the  type  kx^  +  mx  +  n.    Every  trinomial 
of  this  type  which  is  the  product  of  two  binomials,  may  be 
readily  factored  by  an  extension  of  the  method  of  §  63. 
For  example,  to  factor  6  x^  —  11  x  —  Sb^  we  proceed  thus: 
62:2_ii^_35^  1  (36  2;2- 66  a; -210) 

=  i[(6^)2-ll(6.'c)-210] 
=  1  (6  ^  -  2 1)  (6  2;  +  10)  [§  63 

=  (2x-l}(Sx+5). 
The  given  expression  is  first  multiplied  by  6  so  as  to  make 
the  first    term   an   exact  square,   and  the  factor  ^   is  then 
inserted  so  as  to  keep  the  value  unchanged. 

Note  to  the  Teacher.  The  above  method  may,  if  the  teacher  prefers, 
be  replaced  by  the  following ;  in  that  case  §  64  should  follow  §  67. 

Let  6  x^  —  11  a;  —  35  =  (ax  +  b)  (ex  +  d),  wherein  a,  &,  c  and  d  are  to 

be  determined ; 

then  6  aj2  -11  X  -  35  =  acx^  +  (ad  +  bc)x  +  bd, 

whence  —  11  =  ad  +  be  and  6  (  —  35)  =  ac  .  &d  (i.  e. ,  ad  •  be). 

If,  therefore,  we  separate  6 (—35),  le.,  —  210,  into  two  factors  whose 
sum  is  —  11,  we  shall  then  have  found  ad  and  be  ;  these  factors  are  —  21  and 
10,  hence  we  have 

6  a;2  - 11  X  -  35  =  6  x2  +  ( -  21  +  10)  x  -35 
=  6x2-21x+10x-35 
=  3x(2x-7)+5(2x-7) 
=  (2x-7)(3x  +  5). 

When  this  method  is  used  with  young  pupils,  special  care  will  be  needed  to 
keep  the  work  from  becoming  merely  mechanical. 


84  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

EXERCISE  XLII 

1.    In  factoring  3a^  +  13a;-fl4  by  the  method  of  §  64,  what 

multiplier  should  be  used  ?  Why  ?     What   divisor   must  then 
be  used  ?     Why  ? 

Factor  the  following  expressions  and  check  your  work : 

2.  3a^  +  13aj  +  14.  19.    10-19a:4-6a^. 

3.  6a2-lla  +  4.  20.   56  x-\- 15 +  20  a^, 

4.  322  +  ^-10.  21.   Sc^-10cd-3(^. 

5.  4a^  +  16a;  +  15.  22.    -28  +  39s-8s2. 

6.  lOy^-lSy-S.        ^  23.    -  30  i2_;i^9  ^.^5 

7.  9a^  +  7x-2.  24.    12j92_  28^  +  11. 

8.  10ar^  +  a;-2.  25.    16  ar' +  4  a^^^^  _  3Q  ^^4^ 

9.  12i»2_j_43._5^  26.    4:  at)'- 73  abc-\- IS  ac\ 
[Multiply  and  divide  by  3.]  27.     — 14:  y  — 16+15  /. 

10.  lSs^-9s-5.  28.  15a^'*  +  16a;"2/  +  4/. 
[Multiply  and  divide  by  2.]  29.  14  k^  —  27  A;^''  —  20. 

11.  12m2  +  7m-10.  30.  3(a  +  &)' +  10(a  +  &)  -  8. 

12.  20m2-7m-6.  31.  5(c-dy -7(c-d) -6. 

13.  2a^  +  a-55.  32.  15  a.-^^  -  a;^  -  28. 

14.  8^2_^7^-18.  33.  sx'+(7  s-t)x-7t. 

15.  10/  + 7  2/ -12.  34.  cz^+(fc-d)z-fd. 

16.  8^2  +  14  71  —  15.  35.  locP  —  Ikx  —  mx  +  km. 

17.  6i)2-29p  +  35.  36.  6ay^  +  2aby  —  Scy  -be. 

18.  10  62  +  37  6-12.  37.  90  a;^/^^  -  98  a^a??/^  +  8  a^a^y. 

38.  Is  a  product  altered  when  two  of  its  factors  are  changed 
in  sign?  Explain  (cf.  §  18,  also  Ex.  36,  p.  81).  Change  the 
signs  in  each  factor  found  for  Ex.  2  above,  and  thus  write  the 
factors  of  3  x'  + 13  a?  +  14  in  a  new  form.  Similarly,  in  each  of 
Exs.  3-8  write  the  factors  in  a  new  form. 

65.  Squares  of  polynomials.  Cubes  of  binomials.  These 
types  may  be  recognized  by  comparing  them  with  the  for- 
mulas of  §  56  and  §  57. 


04-65]  FACTORING  85 

Thus,  since  the  expression  a^  +  z^  —  4:2/z  +  2xz-\-4^^  —  4ixy 
consists  of  three  square  terms  and  three  double  products, 
it  may  be  the  square  of  a  trinomial.  On  rearranging  its 
terms  thus  :  x^ -\- ^y'^-\-z'^  — 4:xy  -{-^xz  —  -^yz,  and  compar- 
ing with  §  56,  we  see  that  the  given  expression  is  the  square 
oi  x  —  2y  -\-  z. 

Again,  the  expression  12  am^  —  6  a^m  —  8  m^  +  «^,  consisting, 
as  it  does,  of  four  terms,  two  of  which  are  cubes,  may  be  the 
cube  of  a  binomial ;  further  examination  shows  that  it  is  the 
cube  oi  a  —  2  m. 

EXERCISE  XLIII 

1.  Is  a^  —  2a6  +  c^  +  26c  —  2ac  -\-  h^  the  square  of  a  trino- 
mial ?  What  suggests  to  you  that  it  may  be  ?  How  do  you  find 
the  terms  of  this  trinomial  ?  Which  of  them  are  alike  in  sign  ? 
Which  unlike  in  sign  ?     Why  ? 

Factor,  and  check  your  results  as  the  teacher  directs : 

2.  m^  +  w^  -h  s^  +  2  mn  —  2  ms  — 2  ns. 

3.  4:  x^  -\-  y^  -\- 2  yz -{- 4:  xy  +  2-  +  4  xz. 

4.  ^v^-{-2kx  +  x'-Q>kv-&vx-^'k?, 

5.  6ac  +  8  6c  +  9a-  +  c2  +  24a6  +  1662. 

6.  4c2-f 9a2-12ac-fl6ftc-24a6-fl662. 

7.  l-\-2r  —  2m  +  m^  —  2rm  +  r^, 

8.  2lm  —  2ln  +/  —  2lp-^m-  +  l--2mn  —  2mp  +  n^-\-2np. 

9.  If  an  expression  {e.g.,  3  pq- —  (f -\- 2)^  —  3 p^q)  is  the  cube  of 
a  binomial,  how  do  you  find  the  terms  of  this  binomial  ?  By 
what  sign  do  you  connect  them  ?     Illustrate. 

Factor,  and  check  by  §  25 : 

10.  a^-Sa''y-{-3ay'--y\  13.  -  y^  -  12  x'y -^  6  xy^ -\- S  x\ 

11.  m^-f-{-3mf-37nH.  14.  -  27  yh -\- 27  y^-z^ +  9  yz\ 

12.  3a*-f  1-f  3a2  +  a«.  15.  S  -  c^ -12  c' -{-6  c\ 

16.  a;«-2a^  +  10a;2  +  cc*-10aj3  +  25. 

17.  216  - 108  s¥  -f  18  s¥  -  sH\ 

18.  25-^6m^n-10n-\-9m*-30m^-\-nK 


86 


HIGH  SCHOOL   ALGEBRA 


[Ch.  VI 


66.   Factoring  the  type  forms  jr^— /"  and  x"  +/".    By  actual 
division  we  obtain  the  following  results  : 

(^x^  —  a^^  -^  (^x  —  a')  =  X -{■  a, 

(ar^  —  a")  -7-  (x  —  a^  =  x"^  -^  ax -{-  a^^ 

(^x^  —  a^  )  ^  (x  —  (i)  =  x^  4-  ax^  +  oP'x  +  a^, 

(a^  —  a^)  ^{x  —  a)  =  x^-\-ax^  +  a?x^  +  a^x  +  aS  etc. 


I. 


11. 


III. 


(^x^  —  a^)  -^  (x  -{-  a)  =  X  —  a, 
(a;^  —  a^)  -^  {x  +  «)  not  exactly  divisible, 
{x"^  —  rt^)  -h  (x  -]-  a)  =  x^  —  ax^  +  a^x  —  a^ 
(x^  —  a^ )  -V-  (^x -\-  a)  not  exactly  divisible,  etc. 

(^x^  +  «2)  _^  (2;  —  a)  not  exactly  divisible, 
(^x^  4-  rt!^)  -^  (a;  —  a)  not  exactly  divisible, 
(a^*  +  a^)  -r-  (x—  a)  not  exactly  divisible,  etc. 


(x^  +  a^)  -7-  (2:  +  «)  not  exactly  divisible, 

T  V     J  (^^  +  «^  )  ^  (a:  +  «)  =  a;2  —  a:r  4-  ^^ 

(2:*  +  a*)  -f-  (2;  +  a)  not  exactly  divisible, 

(^x*  4-  a^)  -=-  (2:  4-  «)  =  ^*  —  «a;3  4-  oP'x^  —  (^x  4-  a*,  etc. 

These  quotients  illustrate  the  following  principles  (for 
proofs  see  Exs.  17-19,  p.  94): 

(i)  From  I,  x^  —  a^  is  always  exactly/  divisible  hy  x  —  a; 
the  quotient  terms  are  all  positive. 

(ii)  From  II,  x^  —  a^  is  exactly  divisible  by  x -\-  a  only  ivhen 
n  is  even;  the  quotient  terms  are  alternately  positive  and 
negative. 

(iii)    From  III,  x^-\-a^  is  never  exactly  divisible  by  x—  a. 

(iv)  From  lY,  x"-\-a^  is  exactly  divisible  by  x-\-a  only 
ivhen  n  is  odd ;  the  quotient  terms  are  alternately  positive  and 
negative. 

(v)  The  order  of  the  letters  and  exponents  is  the  same  in  all 
the  quotients ;  the  exponent  of  the  first  letter  decreasing.,  and 
that  of  the  second  increasing^  in  passing  toward  the  right. 


66]  FACTORING  87 

EXERCISE  XLIV 

Write  the  following  quotients  by  inspection  and  then  verify 
them  by  actual  division  : 


1.    :^— ^.  7.    •"  '^•'^  .  13.     ■ 

x-\-y 


2. 


x^- 

-r 

X 

-y 

T^- 

-f 

X- 

-y 

a'- 

-b' 

a 

-b 

m^ 

-n« 

m 

+  rt 

u'- 

-v' 

u- 

—  V 

u'- 

-v' 

^jj-jf^  T,      x^y 

x  +  y 
x^  4-  y^ 

x  +  y 


[i.e.,  W^-OTh. 
L  x2  -  y-i       J 


x' 

-f 

(x^r  -  (y^r 

X2  -  y^ 

^10 

+  P 

s^ 

+  ^^ 

^ao 

-?/" 

s' 

-/ 

x'' 

_y2 

x' 

-t 

x'' 

-f' 

9.     '£±^.  14. 

m-\-s 

4.  •.^^.  10.    ^^!±A'. 

a-\-b 

5.  :^^.  11.    (■^7'  +  0/y.  16. 

i^  +  f 

u-\-'\}  (J?  —  &  x?  —  y^ 

18.  In  Exs.  5-11,  above,  express  the  dividend  as  the  product  of 
the  quotient  and  the  divisor. 

19.  Of  which  of  the  following  binomials  is  r  —  s  a  factor : 
,.8  _|_  ^8 .  ^10  _  ^10 .  ^.7  _  ^7 .  ^.11  _|_  gU  9  Answer  the  same  question  for 
the  factor  r  +  s. 

Write  each  of  the  following  as  the  product  of  two  factors : 

20.  m^  —  n^.  26.    x^-^y*^,  32.    l^p'^  —  q'^, 

21.  d^'  +  e^  27.    r^  —  s^.  33.   32.^'^4-l. 

22.  x'^-y'^.  •    28.    2/^  +  8  34.    8-27r^. 

23.  F-Z^.  [Le.,?/3  +  (2)3].  35^    Ho  x' -'^l. 

24.  2/3 +z3.  29.   0^3  +  27.  36_   27v^-64m;«. 

25.  aio  +  &''.  ^^'   8^'-l-  37.   /  +  32a.'io. 
[Cf.Ex.  14.]               31.^^-32.  38.    64r-r^ 

39.  Factor  a^  —  &  in  two  ways :  (1)  by  taking  out  tlie  factor  a  —  c, 
(2)  by  using  §  53  (cf.  Ex.  12,  above)  and  then  refactoring  the  two 
factors  thus  found.  Which  is  the  better  plan  to  use  when  the 
prime  factors  of  a^—  c^  are  sought  ?  Show  that  this  plan  is  advis- 
able in  general,  e.g.^  with  a^  —  y^  and  p^  —  q-^. 


88  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

Resolve  the  following  expressions  into  their  prime  factors : 

40.  x*-y\  45.    aiV^-2/i«.  50.   a^-\-y^ 

41.  a^-b^  46.   64a^-l.  51.   x^^-xy^. 

42.  a^-b^  47.   a«-81.  52.   Sas^'-Saf^ 

43.  m«-l.  48.    SI  a*b*- 16  xy.         53.    64ic«  +  2/^ 

44.  r^^—n^,  49.    a;^  — 2/^  54.   /4-1- 

67.  Factoring  by  rearranging  and  grouping  terms.  A  re- 
arrangement and  grouping  of  the  terms  of  an  expression  will 
often  reveal  a  factor  which  could  not  be  easily  seen  before. 

E.g.,  ax— 3  by-{-bx  —  Say  =  ax-^bx  —  3by  —  Say 

=  x(a-hb)-Sy(a-^b) 
=  (a-\-b)(x-Sy), 

i.e.,  ax  —  3by  +  bx  —  S  ay  =  (a-\-b)  (x  —  3y) . 

Again,  a;(a;  +  4)  —  y(j/  -f-  4)  =  a;^  +  4a;  —  /  —  4?/ 

=  xF  -  y'^ -\- 4:(x  —  y) 
=  (x-y)(x-{-y)-{-4:{x^y) 
=  (x-y){x  +  y-\-4:). 

I.e.,  x(x  +  4:)-y(y  +  4.)  =  (a;-?/)  (a; -h2/  +  4). 

EXERCISE  XLV 
-  Factor  the  following  expressions  and  check  your  work : 

1.  cx  —  cy-{-Sx  —  Sy,  12.  m^  —  n^  —  (m  —  ny. 

2.  ay-^1cx-\-ax-i-ky.  13.  'Sxy(x^-{-y)-^16(x^  +  f). 

3.  p3_p2_^7^_7^  14_  x^ _ xy^  —  ax^ -{- ay\ 

4.  p3_y_7^_^7^  15,   ab-\-bx''  —  x''y'^  —  ay'^. 

5.  ac  +  bd  —  ad—  be.  16.  a^  —  9  a;^  +  4  c^  —  4  ac 

6.  9cy  —  6cx-12mx-\-lSmy.       [i.e.,  (a2-4ac+ 4c2)- 9  a;^]. 

7.  aV-acd-a6c  +  6d.  17.   -  Uk^  -  A9  b^ -\- A  k' -\- 121. 

8.  7mr-3rs  +  21ms-9s2.        18.  ac^  +  M^  -  ad^  -  fec^. 

9.  5a^-«2  +  2-10a;.  19.  1  +  ds - (c^  +  c(^y. 

10.  5a^-hl-x^-5x.  20.   (a +  1)2 -(4 a +  3)2. 

11.  aa^  +  l  +  a  +  ar\  21.    (p^  -  q^  -  (p^  -  pqy. 


00-68]  FACTORING  89 

22.  a^x  +  ahx  +  ac  4-  b^y  +  a6?/  +  &c. 

23.  (ar'  +  6a;  +  9)2-(aj2  +  5a;  +  6)^ 

24.  a^-a2  +  2/=^-62  4.2a;2/-2a6. 

25.  h'-m^-\-10m  +  k^-25-2hk. 

26.  (a;  +  2/)^  +  12(a;  +  2/)-85  (cf-  Exs.  34-35,  p.  83). 

27.  x2  +  4a;^  +  4i/2-f.3a;H-62/  +  2  (cf.  Ex.  26  above). 

28.  4:  x"^  +  10  X  —  6  —  5  a  —  A  ax  -\-  a\ 

29.  Show  that  by  changing  the  signs  of  two  of  them  at  a  time 
the  factors  in  Ex.  10  may  be  written  in  three  different  forms 
(cf.  Ex.  38,  p.  84).     Is  the  same  true  in  Ex.  18  ? 

68.  Factoring  by  means  of  other  devices.  It  often  happens 
that  the  factors  of  an  expression  will  become  apparent  by 
adding  a  certain  number  to,  and  subtracting  the  same  num- 
ber from,  the  given  expression ;  this,  of  course,  leaves  the 
value  of  the  expression  unchanged. 

Ex.  1.  Find  the  factors  ofx^  +  x^  +  l. 

Solution.  If  the  second  term  in  this  expression  were  2x^ 
instead  of  x^,  then  (§  61)  the  expression  could  be  written  (x^  + 1)" ; 
this  suggests  that  x^  be  both  added  and  subtracted,  which  gives 
x*-{-x^-{-l=x'  +  2a^-{-l-x' 

==(^^^l+x)(x^  +  l-x),         [§62 
i.e.,  x*  +  x'i-l=y'  +  x  +  l)(a^-x  +  l). 

EXERCISE  XLVI 

2.  Find  the  factors  of  a*  +  a^b^  +  b\ 
Suggestion.     By  the  method  of  Ex.  1, 

a4  _,.  ^2^2  +  54  =  (^4  _^  2  a-^62  +  54  _  ^252 

=  (a2  +  &2)2_(a5)2. 

3.  Find  the  factors  of  a^  -  4  a;  -  32. 

Suggestion.  Here  the  first  two  terms,  plus  4,  form  an  exact  square  ;  this 
suggests  the  following  arrangement : 

x2_4x-32  =  ic2-4x  +  4-32-4 
=  (x-2)2-36. 


90  EIGH  SCHOOL  ALGEBUA  [Ch.  VI 

4.  What  must  be  added  to  ic^H-3a^+4  to  make  it  an  exact 
square?  What  must  then  be  subtracted  to  leave  the  value  un- 
changed ?     Factor  the  given  expression. 

5.  Can  the  sum  of  two  squares  be  factored  (cf .  §  66)?  Is  a;^  -f  4 
the  sum  of  two  squares  ?     Can  it  be  factored  ? 

6.  What  must  be  added  to  a;^  -f-  4  to  make  it  (a^  +  2)^?  Is  the 
added  term  a  square  ?     Factor  x"^  +  4. 

Factor : 

7:  p^^q^\  16.  4a8-21a46^  +  96^ 

a  x^-f-64  2/^  *         17.  ^x^-lOxY  +  ^y^' 

9.  m^  +  mhi^^-n\  18.  ^  a""  ^2Q>  a'h'' +  25  h\ 

10.  x''^a'x^  +  a\  19.  o? +  2  ab -d''-2hd, 

11.  a^  +  a;y-f/.  20.  4a^  +  81. 

12.  a^  +  6a;4-5.  21.  a^/-h4aj/. 

13.  9  s^ -f  30  si  4- 16  ^^  22.  m^-\-4:m7i\ 

14.  a^6^  +  «'&Vd2  +  c4d^.  23.  a'  +  Sa'-12S. 

15.  9  a;^  +  8  a^2/^  +  4  2/^  24.   5  na;^  —  70  wa;^  +  200  n. 
25.  Find  the  four  factors  oi  x*  i-y' +  z^-2  a^/ -  2 a^2;2 _  2  ^^V. 

69.   General  plan  for  factoring  a  polynomial. 

1.  By  inspection,  find  and  remove  all  monomial  factors. 

2.  By  comparison  with  type  forms,  by  rearrangement  and 
grouping  of  terms,  or  by  some  other  device,  separate  the 
resulting  polynomial  factor  into  two  factors. 

3.  Then,  if  possible,  separate  each  of  these  factors  into 
two  others,  and  so  continue  until  all  factors  are  prime. 

Note.  By  the  above  plan  the  simpler  expressions  can  usually  be  factored. 
For  determining  the  binomial  factors  of  longer  polynomials,  see  §  71. 

EXERCISE  XLVII 
Factor : 

1.  4aa^-4a/.  4.   s^  + 16  s^ -]- 15 s. 

2.  49A;-fc3^  5.   x*-Sx^-^15x\ 

3.  fp—p\  6.   mV  +  my. 


68-69]  FACTORING  91 

7.  tv  -  nw  -  nv -\- tw.  14.  k* —  17  k^ -\-16. 

.     8.  v^   -7?;--2i;  +  14.  15.  Sx-^ —  lOxy +  37f, 

9.  1  -  24  .s -h  144  si  16.  49mV  +  42wms-h9n2. 

10.  a''b'-4.abhj-^4:by.  17.  c-  + a;2-2  ex- 1. 

11.  p^  —  Sr\  18.  &*  +  6y  +  2/^ 

12.  1-2/'.  19.  2u^-Uu^  +  70-107L 

13.  c2-5c-14.  20.  4.c'-25a^-\-b^-4:bc. 

21.  Give  two  methods  for  checking  an  exercise  in  factoring. 
Illustrate,  using  Ex.  15  above. 

Factoi:  and  check  as  the  teacher  directs : 

22.  71^  —  1.  38.  a^x^  —  oc^, 

23.  q*p  —  t^*.  39.  s^^  —  t^^. 

24.  216  +  2/3.  40.  Tr'-.QOT. 

25.  a«  +  4.  41.  a^^  +  l. 

26.  3(a;-2/)^-27.  42.  (a -6)^-03. 

27.  rsv^  —  ar^st  —  4:  cr^ .  43.  /c^^  +  4p^ 

28.  x^-\-ax-ay-yx.  44.  12  4-s(^2_4)  _3  ^2^ 

29.  m(d2-3)+d2_3^  45;  x^''^^-{-2x''-'by  +  by, 

30.  m^"  —  4:  m^b -\- 4.  b\  46.  m^  — 1  —  3  m(m  —  1). 

31.  fc(Z2-4)-/2  +  4.  47.  3a2p-28rg-21ri94-4aV 

32.  mV  +  4-5mV.  48.  (5a  +  2/)2-7(5  a +?/)  +  10. 

33.  7(a;  +  a)-ll(a;2-tt2).  49.  8 -12  mn  + 6  mV-m%l 

34.  y^-y^L  50.  m2  +  6m/i-16.T2/  +  9nl 

35.  -a;y-;2(3  2/-a;-3  2;).  51.  (a? -1/)^- 2  2/  +  2  aj+l. 

36.  x^  —  y^  —  3xy  +  3a^y\  52.  m^n^  +  2  mVrV  +  mV W. 

37.  6x''  +  12a^~lS.  53.  (a2  +  5a4-4)2-(a2-5a-6)l 

54.  a;2-2x2/  +  l+/  +  2(a;-2/). 

55.  (c-3)«-l-3(c-3)2  +  3(c-3). 

56.  m^~2mn-[-n^-s--{-2st-f. 

57.  {c'-2cd-\-dy-(Sc'-cd-2d'y. 

58.  «2  +  9/  +  2522_6iC2/-10^^  +  302/«. 

59.  2(a262_aV-6V)  +  a^  +  6*  +  c*. 

HIGH  SCH.  ALG.  —  7 


92  HIGH  SCHOOL  ALGEBRA  [Cii.  VI 

60.  a¥  ^  yiy.  _  ^2g  _^  ^2^  _|_  52^2  _  ^2g^ 

61.  a2_2a6  +  6'-2ac  +  26c  +  c2-2ad  +  26d  +  2cd  +  d2. 

62.  iB2-9a;-|-14  =  (aj-7).(  ?  )  =  (7-a;).(  ?  ). 

63.  c3-r3=(c-r).(  ?  )  =  (r-c).(  ?  ). 

64.  7'3_36r  =  r(r+6).(  ?  )  =  -r(r  +  6).(  ?  ). 

65.  Write  the  four  factors  of  a;*  — 10  a^  +  9  in  seven  different 
ways  (cf.  Ex.  29,  p.  89). 

70*  Remainder  theorem.  In  Ex.  53,  p.  52,  it  was  seen  that 
if  oc^ -^  3  X -\- 1  is  divided  by  a;  —  a  the  remainder  is  a^  +  3  a  4- 1 ; 
i.e.,  the  remainder  is  what  the  dividend  would  become  if  a  were  sub- 
stituted for  X.     (Cf.  also  Ex.  52,  p.  52.) 

And  this  relation  between  dividend  and  remainder  is  not  acci- 
dental ;  it  is  true  for  all  such  expressions.     For,  let 

Ax""  +  Bx""-^  +  Cx""-^  H [-Hx  -\-K  • 

be  any  polynomial  in  x,  let  it  be  divided  by  x  —  a,  and  let  Q  and 
Rj  respectively,  represent  the  quotient  and  remainder;  then 

Ax^  +  Baf-^  +  Cx^-'-^  ...j^Hx  +  K=  Q(x-a)  +  R. 
Moreover,  since   the   second  member   of  this   equation,  when 
multiplied  out,  must  be  exactly  like  the  first,  therefore  this  equar 
tion  is  true  for  all  values  that  may  be  assigned  to  x;  but  if  the 
value  a  be  given  to  x,  the  equation  becomes 

Aa'^  +  5a"-i  +  Ca^-^  +  - -- +  Ha -\- K=:  R,f 
hence,  in   every  such  division,  the  remainder  may  be  obtained  by 
simply  substituting  a  for  x  in  the  dividend. 

71.*   Application    of   the   remainder   theorem   to   factoring.      By 

means  of  the  remainder  theorem  (S  70),  and  without  actually 
performing  the  division,  write  down  the  remainder  resulting 
from  dividing  a^— 3a^  +  3a;-f2  by  a;  —  a.  Also  write  the  re- 
mainder when  aj^  —  3a^-f-3a;  —  2  is  divided  by  a;  —  2.     What  is 

*  Articles  70  and  71,  with  Exercise  XLVIIJ,  may,  if  the  teacher  prefers, 
be  omitted  till  the  subject  is  reviewed. 

t  Since,  in  that  case,  Q(x  —  a)  becomes  Q'(a  —  a),  i.e.,  zero;  and  M  is 
the  same  as  before  substituting,  since  it  does  not  contain  x. 


60-71]  FACTORING  93 

the  value  of  this  last  remainder?  Does  this  show  that  a;— 2  is 
a  factor  of  y? -'d  x^ +  Sx-2'! 

Binomial  factors  of  many  polynomials  may  be  found  in  this 
way,  for,  from  §  70,  it  follows  that  if 

^a"  +  BoT-"^  +  (7a"-2  -\ +Ha  +  K^  0, 

then,  and  then  only,  is  Ax""  +  Bx""-^  +  Ca;"-^  -\ h  ^a;  +  /f  ex- 
actly divisible  by  ic  —  a ;  for  in  that  case,  and  in  that  case  only, 
is  the  remainder  zero. 

Thus,  we  know  that  a?  —  3  is  a  factor  of  y?  —  2y?  —  ^:X  -\-Z 
because  3^  — 2-3-— 4.34-3  =  0;  and  a;-f-l,  ?.e.,  a?— (— 1),  is  a 
factor  of  a;2  +  7  a;  -f  6  because  (- 1)2  ^_  7(_  i)  +  6  =  0. 

Again,  if  a;  —  a  is  a  factor  of  a^  —  a;^  —  2  a;  +  8,  then  a  is  a  factor 
of  8 ;  hence,  in  seeking  such  factors  of  a?^  —  a;^  —  2  a;  -f  8  we  need 
try  only  1,  —  1,  2,  —  2,  4,  —  4,  8,  and  —  8  in  place  of  a. 

When,  by  any  process  whatever,  any  factor  of  an  expression 
has  been  discovered,  this  factor  may  be  divided  out;  the  remain- 
ing factors  may  then  be  more  easily  found. 

EXERCISE  XLVIII 

1.  If  a;^-f  6a^— 12  a;  +  5  is  divided  by  x  —  a,  what  is  the 
remainder  ?  Without  performing  the  division,  find  the  re- 
mainder when  the  divisor  is  x  — 2;  also  when  it  is  a;  — 1  and 
when  it  is  x-\-l.  Which  of  these  divisors  is  a  factor  of  the 
given  expression  ? 

2.  If  the  expression  x^  —  3a^  —  a;-f-3  has  a  factor  of  the  form 
X—  a,  what  are  the  four  possible  values  of  a ?  Find  all  the 
binomial  factors  ofa;^  —  3ar^  —  a;-f3. 

By  the  above  method,  factor  the  following  expressions : 

3.  a^_7a;-f6.  7.   T<? -\-4.k^ -Ilk -30. 

4.  a;3-9a^4-23a;-15.  8.   w^  -  15  w;^  +  10  lo -f  24. 

5.  a.-3  +  14ar-f-35a;-f-22.  9.   a^-\-l  a" -\-2  a- ^0. 

6.  a.-3-lla;^-f-31a^-21.  10.    c^-S  c^- 29  c  +  105. 

11.  .^•*  -  a.-*^  -  7  a^  +  a;  4- 6. 

12.  /-102/'  +  402/^-80/-f 80y-32. 


94  HIGH  SCHOOL  ALGEBRA  [Ch.  VI 

13.  If  X  —  A:  is  a  factor  of  any  given  expression,  what  does  the 
value  of  that  expression  become  when  x  =  k  ^  Why  ?  If  any 
given  expression  becomes  zero  when  x  =  k,  is  x  —  k  a  factor  of 
the  expression  ?     Why  ? 

14.  By  means  of  the  remainder  theorem  show  that  a—b,  b  —  c, 
and  G  —  a  are  factors  of  a(b^  —  c'-^)  +  b(c^  —  a^  +  c{a^  —  b^). 

15.  Write  the  remainder  when  (2x  —  Saf  +  (Sx  —  ay  is  divided 
by  ic  —  a ;  also  the  remainder  when  (x—  y  -\-zy  —  'if-\-x^  is  divided 
by  a;  —  2/ ;  and  hj  x-{-y,  that  is,  hj  x—{  —  y). 

16.  What  is  the  remainder  when  or"*— ctMs  divided  hjx  —  a? 
Why?  Write  the  remainder  when  x'' -\- a^  is  divided  by  a;  — a; 
when   ic^  +  a^   is  divided  by  x  -\-  a. 

17.  By  means  of  the  remainder  theorem,  show  that  x^  —  a"  is 
exactly  divisible  by  a;  —  a ;  also  that  x''  +  a"  is  not  exactly  divis- 
ible by  £c  —  a  (cf.  §  Q^Q). 

18.  By  means  of  the  remainder  theorem,  show  that  x""  —  a"*  is 
exactly  divisible  by  x  +  a  only  when  n  is  an  even  positive  integer. 

19.  By  means  of  the  remainder  theorem,  show  that  x"^  +  a"*  is 
exactly  divisible  hy  x-\-a  only  when  n  is  an  odd  positive  integer. 

72.  Solving  equations  by  factoring.  Factoring  greatly 
simplifies  the  solution  of  certain  kinds  of  equations.  The 
following  examples  illustrate  the  procedure. 

Ex.  1.  Given  ic^  — 5a;  +  6  =  0;  to  find  its  roots,  i.e.,  to  find 
those  values  of  x  for  which  this  equation  is  satisfied  (cf.  §  45). 

Solution.  By  §  63  the  first  member  of  this  equation  is  the 
product  of  ic  —  3  and  a;  —  2 ;  hence  the  equation  may  be  written 
thus: 

(a;-2)(a;-3)=0. 

Now  this  equation  is  satisfied  if  either 

x-2  =  0  or  a?-3  =  0,  [§  41 

i.e.,  if  either  a?  =  2  or         a;  =  3. 

On  substitution  these  values  are  found  to  check;  they  are, 
therefore,  the  roots  of  the  given  equation. 


71-72]  FACTORING  95 

Ex.  2.     Given  oc^  =  8  x -\- 4: ;  to  find  its  roots. 

Solution.     On  transposing,  tliis  equation  becomes 
x^-3x~4:={), 
i.e.,  (x_4)(a;  +  l)=0;  [§63 

hence  either  ic  —  4  =  0  or  a^  +  1  =  0, 

i.e.,  x  =  4:         or£c  =  — 1; 

and  these  numbers  check,  therefore  the  roots  are  4  and  —  1. 

Ex.  3.     Solve  the  equation  6  a?^  — 11  cc  =  35. 

Solution.  On  transposing  and  factoring  (§  64),  this  equation 
becomes 

(3x+5)(;2x-7)  =  0', 
hence  305  + 5  =  0  or  2a;  — 7  =  0;    . 

therefore  the  roots  are  —  |  and  J. 

Remark.  Since  the  roots  of  the  equation  (x  —  a)(x—b)=:0 
are  a  and  b,  therefore  an  equation  which  shall  have  any  given 
numbers  as  roots  may  be  immediately  written  down;  thus  the 
equation  whose  roots  are  3  and  8  is 

(x-3){x-S)  =  0,     I.e.,  a^- 11  a; +  24  =  0. 
Similarly,  the  equation  whose  roots  are  2,  —  1,  and  5  is 
(x-2)(x-\-l)(x-5)  =  0,     i.e.,a^-(jx'-{-Sx-\-10  =  0. 

EXERCISE  XLIX 

4.  What  is  meant  by  a  root  of  an  equation  (cf .  §  45)  ?  May 
an  equation  have  more  than  one  root  ? 

5.  What  values  of  x  satisfy  the  equation  (x—2)(x—3')=0? 
Can  any  values  of  x  other  than  2  or  3  sat^*  sfy  this  equation  ? 
Explain.     How  many  roots,  then,  has  this  equation  ? 

Solve  the  following  equations  by  factoring,  and  check  the  roots : 

6.  2/--62/  +  5  =  0.  11.    ^2-4  =  0. 

7.  x^-4:x-21  =  0.  12.   m2-36  =  0. 

8.  ^-13s-\-A0  =  0.  13.   /-7?/  =  0. 

9.  a;2-2a;  =  15.  14.   c2  +  22c  =  -121. 
10.   A;2  +  47(;  =  45.  15.   '^^-3^-50  =  38. 


96  HIGH  SCHOOL   ALGEBRA  [Cii.  VI 

16.  32/-  +  ?/-, 10  =  0.  22.  a;--3aa;-54a2  =  0. 

17.  6ar^  — a;=l.  23.  s^  —  (c  +  d)s -f  cd  =  0. 

18.  4y2_27  =  12'y.  24.  8a^  +  10.T  =  3. 

19.  82/2  +  15  =  -26y.  25.  36  =  -x^+13a:2^ 

20.  5if2— 7a;  =  0.  26.  ^-\-x~  —  x  =  l. 

21.  1222  =  _42.  27.  2a^  +  5.^  =  2a;  +  5. 

28.  What  are  the  roots  of  {x  -l){x -2){x  -^2)  =  0?  Explain. 
Determine  by  inspection  the  roots  of  (a;+l)(3  a;— 2)  =  0. 

29.  Determine  by  inspection  the  roots  of : 

(1)  (5a;-3)(a;-l)=0. 

(2)  (2/-f)(22/  +  9)  =  0. 

(3)  m(3  m  + 1)(4  m  -  3)  =  0. 

(4)  («-a)(2aj-lla)(4a;  +  5a)  =  0. 

30.  Write  an  equation  whose  roots  are  5  and  2.  Also  one 
whose  roots  are  3,  1,  and  7. 

31.  Write  the  equations  whose  roots  are :  1  and  —  5 ;  f  and  6 ; 
a  and  65  3,  —  1,  and  o ;  a,  —a,  and  2a;  1,  2,  3,  and  4. 

The  following  problems  lead  to  equations  whose  roots  may  be 
found  by  factoring.     Solve  and  check  each  problem. 

32.  Find  a  number  such  that  if  3  and  5  are  subtracted  from  it 
in  turn,  the  product  of  the  two  remainders  is  24.  How  many 
solutions  has  this  problem  ?     Explain. 

33.  The  sum  of  two  numbers  is  12,  and  the  square  of  the 
larger  is  1  less  than  10  times  the  smaller.  Find  the  numbers  (cf. 
Ex.  18,  p.  6).  • 

34.  Tlie  difference  between  two  numbers  is  2,  and  the  sum  of 
their  squares  is  130.     What  are  these  numbers  ? 

35.  One  side  of  a  rectangle  is  3  feet  longer  than  the  other.  If 
the  longer  side  be  diminished  by  1  foot  and  the  shorter  side  in- 
creased by  1  foot,  the  area  of  the  rectangle  will  then  be  30  square 
feet.     How  long  is  this  rectangle  ? 

Note.  The  equation  of  this  problem  has  two  roots,  one  positive  and  one 
negative  ;  but  only  the  positive  root  will  satisfy  the  problem  itself,  for  it  is 
implied  that  the  dimensions  of  the  rectangle  are  positive. 


72]   .  FACTORING  97 

36.  How  may  $  128  be  divided  equally  among  a  certain  num- 
ber of  persons  so  that  the  number  of  dollars  received  by  each 
person  shall  exceed  the  number  of  persons  by  8  ? 

37.  The  senior  class  of  a  certain  school  present  the  school  with 
a  picture  whose  cost  is  $12.  If  each  senior  contributes  3  times 
as  many  cents  as  there  are  members  in  the  class,  how  large  is  the 
class  ?     How  much  does  each  member  pay  ? 

38.  A  rectangular  orchard  contains  2800  trees,  and  the  number 
of  trees  in  a  row  is  10  less  than  twice  the  number  of  rows.  How 
many  trees  are  there  in  a  row  ? 

39.  If  the  dimensions  of  a  certain  rectangular  box  which  con- 
tains 120  cubic  inches  were  increased  by  2,  3,  and  4  inches, 
respectively,  the  new  box  would  be  cubical  in  form.  Find  the 
dimensions  of  this  box  (cf.  §  71). 


CHAPTER   VII 

HIGHEST   COMMON   FACTOR  — LOWEST    COMMON    MULTIPLE 

I.  HIGHEST   COMMON  FACTOR 

73.  Definitions.  A  factor  of  each  of  two  or  more  algebraic 
expressions  (or  numbers)  is  called  a  common  factor  of  these 
expressions.  The  highest  common  factor  (H.  C.  F.)  of  two 
or  more  expressions  is  the  product  of  all  the  prime  factors 
(§  58)  that  are  common  to  these  expressions;  it  is,  therefore, 
the  factor  of  highest  degree  common  to  the  given  expressions. 

Thus,  the  H.  C.  F.  of  9  a%V  and  6  abh"^  is  3  ah\  because 
when  this  factor  is  removed  from  the  given  expressions  they 
have  no  common  factor  left. 

So,  too,  2x(^a-  1)2  is  the  H.  C.  F.  of  6  aV  (a  -  1)*  and 
Sx(a-iy(is-ty, 

Two  or  more  algebraic  expressions  which  have  no  common 
factor  except  unity  are  said  to  be  prime  to  each  other. 

74.  Highest  common  factor  of  two  or  more  monomials.    The 

H.  C.  F.  of  two  or  more  monomials  may,  obviously,  always 
be  found  by  inspection. 

E.g.,  to  find  the  H.  C.  F.  of  12  a^b^xy,  6  a6V,  and  9  ab^x\ 
Inspection  shows  that  3,  a,  b^,  and  x  are  the  only  factors  com- 
mon to  the  given    monomials;  hence  the  H.  C.  F.  of  these  mo- 
nomials is  3  ab^x. 

A  rule  for  writing  down  the  H  .C.  F.  of  several  monomial 
expressions  may  be  formulated  thus :  To  the  H.  C.  F.  of  the 
numerical  coefficients  annex  those  letters  that  are  common  to 
the  given  monomials,  and  give  to  each  of  these  letters  the  lowest 
exponent  which  it  has  in  any  of  the  mono^nials. 

98 


73-75]  HIGHEST  COMMON  FACTOR  99 

75.   H.  C.  F.  of  polynomials  whose  factors  are  known.     By 

first  writing  any  given  polynomials  in  their  factored  forms 
their  H.  C.  F.  may  be  found  by  inspection. 

For  example,  to  find  the  H.  C.  F.  of  4  ax^  —  20  ax -\- 24:  a  and 
6  abx^  +  24  abx  — 126  ab,  we  write  : 

4:ax'-20ax-\-24:a  =  4.a{x-2)(x-3),  [§  63 

and  6  aba^ +  24:abx-126  ab  =  ^  ab(x-\-7)  (x-S)-, 

hence  their  H.  C.  F.  is  2  a  (x—3). 

EXERCISE  L 

Find  the  H.  C.  F.  of  each  of  the  following  sets  of  expressions  : 

1.  3  a'b' Sind  6  ab\ 

2.  15  afy%  24  xY,  and  18  xSj. 

3.  16  x'fz',  52  yh%  and'39  x'f- 

4.  195  a*6V  and  260  a'bc\ 

5.  96  ?/V,  100  ?/V,  and  56  t/V. 

6.  104  x'^y^^'z^'  and  364  x^'^f^'z^'. 

7.  (c  +  d)\c  -  d)  and  (c  +  d)(c  -  d)\ 

a   6(c  +  d)2(c-d)2and  15(c-(^)2(c+c?). 
9.   24  a^x  (y  -  zf  and  56  a'^b^  (y  -  z)*. 

10.  a^  —  b^,  a(a  —  b),  and  a^  —  2ab-\-  6^ 

11.  a;2_^7a;  +  10and  a.-2  +  12x  +  20. 

12.  'm?  —m—  12  and  di?  —  4  m  —  21. 

13.  15  (2/2;  —  z)  and  35  (/s;  —  yz). 

14.  Is  —  (tt  —  b),  I.e.,  6  —  a,  a  common  factor  of  the  expressions 
in  Ex.  10  (cf.  Ex.  36,  p.  81)?  May  we  then  call  the  H.  C.  F. 
of  these  expressions  either  a  —  6  or  6  —  a? 

15.  Show  that  the  H.  C.  F.  of  rn?  —  mn  and  n^  —  mn  is  either 
m  —  71  or  71  —  m. 

In  each  of  Exs.  16-25  find  two  forms  of  the  H.  C.  F. : 

16.  1^  —  s^  and  ^  —  rl 

17.  5  a  —  as  and  3  s'  —  75. 

18.  p^  -  125  and  p^  -  10  p^  +  25  p. 


100  HIGH  SCHOOL  ALGEBRA  [Ch.  VII 

19.  a^  4-  a^  and  3  a^  +  3  a^x  —  5  aoiy^  —  5  a.*^. 

20.  28^2-17?! -3  and  4n2  +  5/i-6. 

21.  5-19 A;-4A;2  and  A;2^2A;-15. 

22.  a^a;  —  x  —  y-\-  a^y  and  a^a?  +  4  a^a;  —  5x. 

23.  12  a6^a;  +  4  a6^a;^  —  40  aft^,  18  a^mx^—  54  a%a;  +  36  a^m,  and 
6  aV?/  —  6  a^xy  —  12  a^y. 

24.  ?^v  —  w^,  u^—  5v  -\-5u  —  uVj  and  3  w^  —  10  iii;  +  7  v^. 

25.  15  aV  4- 15  a'6V  + 15  ft^aj^  and  3  {a'  -  ab^  +  ^>')- 

Find  the  H.  C.  F.  of  each  of  the  following  sets  of  expressions: 

26.  2a^-a;-3  and2a^  +  llar^-a;-30. 

Suggestion.  Find  the  factors  of  2  x^  —  x  —  3  and  determine  by  trial  which 
of  these  are  factors  of  2  cc^  +  11  a;^  _  ^  —  30  also.  This  plan  may  be  used 
whenever  any  one  of  a  given  set  of  expressions  is  easily  factored. 

27.  (a;+3)(aj2-4)  anda;^-h4a^  +  2a.'2-aj  +  6. 

28.  a^  +  1,  Sa^-4.a^-\-4.a-l,  and  2  a^  + a^- ^  +  3. 

29.  b^-S,  b'^b'  +  2b-4.,  ^ndb'-\-2b'-b'-10b-20. 

30.  Of  what  is  the  H.  C.  F.  of  two  or  more  expressions  com- 
posed ?  State  a  rule  for  finding  the  H.  C.  F.  of  two  or  more  ex- 
pressions which  may  easily  be  separated  into  their  prime  factors. 

31.  Is  the  H.  C.  F.  as  above  defined  the  same  as  the  greatest 
common  divisor  (G.  C.  D.)  in  the  arithmetical  sense  ?  What  is 
the  H.  C.  F.  of  a^(a;  - 1)^  and  xia^-l)?  Is  this  also  the  G.  C.  D. 
of  these  expressions  for  all  values  of  x?     Try  a;  =  3,  also  a;  =  4. 

76.^  H.  C.  F.  of  polynomials  neither  of  which  is  easily  factored. 
The  H.C.F.  of  two  or  more  polynomials  can  always  be  found  by 
what  is  known  as  the  Euclidean  (division)  process.  This  process 
is  essentially  the  same  as  that  used  in  arithmetic  to  find  the 
G.  C.  D.  of  two  numbers. 

The  steps  in  the  arithmetical  process  are :  (1)  Divide  the  larger 
number  by  the  smaller ;  (2)  if  there  is  a  remainder,  divide  the 
smaller  number  [I'.e.,  the  divisor  in  step  (1)]  by  this  remainder; 

*  Articles  76,  77,  and  78,  with  Exercises  LI  and  LIT,  may,  if  the  teacher 
prefers,  be  omitted  till'  the  subject  is  reviewed. 


:(J] 


HIGHEST  COMMON  FACTOR 


101 


(3)  divide  the  remainder  in  (1)  by  the'  remainder  in  (2);  (4)  so 
continue,  dividing  each  remainder  by  the  one  following,  until 
there  is  no  remainder;  (5)  the  last  divisor  is  the  G.C.D.  sought. 


This  work  may  be  more  compactly 
arranged  thus : 


2639 
2866 


Thus,  to  find  the  G.  C.  D. 
of  1183  and  2639. 
1183)2639(2 
2366 

273)1183(4 
1092 
91)273(3 
273 
0 

The  last  divisor,  91,  is  the  G.  C.  D.  of  the  given  numbers. 
Similarly,  the  H.  C.  F.  of  a:*  +  3  a;^  +  2  ic-^  -  x  -  5  and  x^  +  x^  -  2  may  be 
found  thus  : 


1183 

1092 
91 


QUOTIENTS 


273 

273 


QUOTIENTS 


x*  +  3  x3  +  2  x"^ 
x*+     x3 


X 

2x 


2  x3  +  2  x2  +     X 
2  x3  +  2  x2 


X-  1 


x  +  2 

a;3  4x2-2 

X3-X2 

2x2-2 

2  x2  -  2  X 

x2  +  2  X  +  2 

2x  -2 

2x  -2 

0 

Hence  x  —  1,  the  last  divisor,  is  the  H.  C.  F.  of  the  given  polynomials. 


EXERCISE  LI 

By  the  above  method,  find  the  H.  C.  F.  of  each  of  the  following 
pairs  of  expressions : 

1.  a^  +  5i»4-6and4a^  +  21a;2  +  30a;  +  8. 

2.  6a2-13a-5andl8a3-51a2  +  13a  +  5. 

3.  5  m'''  —  2  m  —  3  and  15  m^  —  6  m^  —  4  m  -f-  3. 

4.  c3_2c2-2c-3  andc^-c3-3c2^4c-2. 

5.  12x'-Sa^-55a^-2x-^5and6a^-x'-29x-15. 

6.  lSx^-\-75a^-{-17x'-2Sx-lSand6a^-}-2Sx^-3x-10. 

7.  S0y'  +  16y*  +  16f-Sy^-3y-2a,nd20f  +  4:y'-y-3. 

8.  4.k'-\-201(^-10k'-^SJc-\-S5sind21i^  +  llk'-25, 

9.  5n*-10n3  +  lln2-67i-hl  and 


10 


5n*-7n^  +  ldn'-Un-{'2. 


102 


HIGH  SCHOOL  ALGEBHA 


[Ch.  VII 


11*  Proof  of  principle  involved  in  §  76  (see  footnote,  p.  100). 
The  success  of  the  method  employed  in  §  70  is  due  to  the  follow- 
ing considerations : 

Let  A  and  B  represent  any  two  polynomials  in  ic,  the  degree  of 
A  being  at  least  as  high  as  that  of  B,  and  let  q  and  R  represent 
the  quotient  and  remainder  respectively,  when  A  is  divided  by  B\ 
then  A  =  qB^R.  [Ex.  20,  p.  50 

This  equation  shows  that:  (1)  every  divisor  common  to  B  and 
II  is  a  divisor  of  A  also  (why  ?),  and  (2)  every  divisor  common  to 
A  and  ^  is  a  divisor  of  H  also  (why?);  hence  the  H.  C.F.  of  B 
and  R  is  the  same  as  that  of  A  and  B. 

If  now  B  is  divided  by  R,  giving  p  and  M  as  quotient  and 
remainder  respectively,  then,  by  reasoning  as  above,  we  see  that 
the  H.  C.  F.  of  M  and  7^  is  the  same  as  that  of  B  and  R,  and 
therefore  the  same  as  that  of  A  and  B. 

Suppose  now  that  this  series  of  divisions  is  continued ;  then,  by 
the  above  reasoning,  the  H.  C.  F  of  ^  and  B  is  the  same  as  that  of 
any  ttvo  successive  remainders. 

If  now  the  last  one  of  this  series  of  divisions  is  exact,  i.e.,  if  the 
final  remainder  is  zero,  then  the  H.  C.  F.  of  the  two  preceding 
remainders  is  the  last  divisor  itself ;  hence  the  last  divisor  is  the 
H.C.F.  of  A  and  B,  which  was  to  be  found. 

Remark,  The  H.  C.  F.  of  two  expressions  is  evidently  not 
altered  by  multiplying  (or  dividing)  either  of  them  by  any  num- 
ber which  is  not  a  factor  of  the  other ;  this  fact  enables  us  to 
avoid  fractional  coefficients  in  the  division  process. 

Thus,  to  find  the  H.  C.  F.  of  3  x^  -|-  8  x'^  -h  3  x  -  2  and  a;^  -  2  a;^  +  x  +  4  : 


3x3  +  8x2  +  3x-    2 
3x3-6x2  4- 3x  + 12 


14)14x2-14 


x2-l 

x2-f-X 


-x-1 
-x-1 


x-2 


x=^  -  2  x2  4-  X  -f-  4 


-  2  x2  -}-  2  X  +  4 

-  2  x2  -H  2 

2)2x4- 


X4-1 


Before  beginning 
the  second  division 
the  factor  14  is  sup- 
pressed (see  Remark 
above),  and  later  2 
is  suppressed  also ; 
fractional  coeffi- 
cients are  thus 
avoided. 


Hence  x  4-  1,  the  last  divisor,  is  the  H.  C.  F.  of  the  given  expressions. 


77-78] 


HIGHEST  COMMON  FACTOR 


103 


As  a  further  illustration,  let  us  find  the  11.  C.  F.  of 

x4  +  4  x^  +  2  x2  -  X  4-  6  and  2  x^  +  9  x^  +  7  x 


6. 


xH4x3  +  2x2-x+6 

2 

X,   +1 
2x-l 

2x3+  9^2_^   7a^_6    i 
2x3+10x2+12x 

2xH8xa  +  4x2-2x+12 
2x4+9x3  +  7  x2-6x 

-x-i-  5x-6 
-x2-  5x-6 

-  x3-3x2+4x+12 

-2 

0 

2x3+6x2-  8x-24 
2x3+9x2+  7x-  6 

_3)_3x2-15x-18 

x2+  5x+  6 

Before  beginning 
the  division  the  fac- 
tor 2  is  introduced 
so  as  to  avoid  frac- 
tional coefficients  in 
the  quotient ;   later 

—  2  is  introduced 
for  the  same  pur- 
pose ;     and    finally 

—  3  is  rejected. 
Hence  x2  +  5  x  +  0,  the  last  divisor,  is  the  H.  C.  F.  of  the  given  expressions. 

78.*  Supplementary  to  §  77  (see  footnote,  p.  100).  (i)  If  the 
polynomials  whose  H.  C.  F.  is  sought  contain  monomial  factors, 
these  should  be  set  aside  before  the  division  process  is  begun. 
Monomial  factors  that  are  common  to  the  given  polynomials  must, 
of  course,  be  reserved ;  all  others  may  be  rejected. 

(ii)  The  H.  C.  F.  of  three  or  more  polynomials  is  found  by  first 
finding  the  H.  C.  F.  of  any  two  of  them,  then  the  H.  C.  F.  of  that 
result  and  the  third  polynomial,  and  so  on  until  all  the  poly- 
nomials have  been  used. 

(iii)  To  find  whether  polynomials  which  involve  more  than  one 
letter  have  a  common  factor  containing  any  particular  one  of  these 
letters,  they  need  only  be  arranged  according  to  powers  of  that 
letter,  and  divided  as  already  described.  By  a  repetition  of  this 
process  all  the  common  factors  of  such  polynomials,  and  hence, 
their  H.  C.  F.,  may  be  found. 

[For  fuller  discussion  of  H.  C.  F.  see  El.  Alg.  pp.  116-121.] 


EXERCISE  Lll 
Find,  by  the  Euclidean  method,  the  H.  C.  F.  of : 

1.  ar^-3a;2^3^_j^  andaj4-2ar^+-2a;2-2ic  +  l. 

2.  c^  +  4c3-12c2-f  c  + Gaud  c^- 0^-2  C2+-C  +  1. 

3.  5z^  +  lSz^-3Sz-\-10sind2z^-{-5z^-22z-\-W. 

4.  2ii^-{-Sx'-\-2x-2sindx'  +  2a^-{-x'-2x-2. 

5.  r^-2r^  +  2r2-4andr^+-2r^-7'3-2. 


104  HIGH  SCHOOL  ALGEBRA  [Ch.  Vll 

6.  1  —  4  m^  4-  3  m''  and  1  —  5  m^  +  4  m'*  +  m  —  m^. 

7.  63  +  A:^  -  9  A;  -  7  A;2  and  40  A;  +  A:^  -  5  Ar^  +  111  -  23  fe2^ 

8.  ar^- 4  a;3_  2  a^  _  8  H- a;4  and  2  a^+ 9  ic^-a^- 4  ar^  + 14  a; -16. 

9.  8a^-22a^  +  17a;-3and6x=^-17a^  +  14aj-3. 

10.  2a;2-3a;-35anda;4  +  14a!--9cc3  +  35a;-25. 

11.  What  is  meant  by  the  H.  C.  F.  of  two  expressions  A  and  B  ? 
If  a  is  not  a  factor  of  A,  how  does  the  H.  C.  F.  of  A  and  a  •  B 
compare  with  the  H.  C.  F.  of  ^  and  B  ?     Explain. 

12.  If  a  is  a  factor  of  A,  but  not  of  B,  how  does  the  H.  C.  F. 
of  A  and  a  •  B  compare  with  the  H.  C.  F.  of  ^  and  B?  In  intro- 
ducing and  suppressing  factors  during  the  process  of  division 
(§  77),  what  precaution  must  be  exercised,  and  why  ? 

Find  the  H.  C.  F.  of  the  following  expressions : 

13.  m*  —  3  m^  + 1  and  m^  —  2  m  -  2  —  m*^  —  m^  +  2  m^. 

14.  a'  +  2a'-5a'-10anda'-{-a^-a'-2a-2. 

15.  a^-4x'-2-\-3x-3i^-{-Ba^Siudx-\-2s(^-{-2-5x'. 

16.  s^-2s^-2s3-lls=^-s-15  and 

2s^-7s^  +  4s3-15s2  +  s-10. 

17.  a;^  +  3  ar^  -  2  «2  _  6  a;  and  4  iK^  -  a^  +  a^  +  4  a;^  -  12  +  4  a^. 

18.  21  ax  — 17  ax^  —  5  aar^  4-  ax^  and  5  aa^  —  34  aa^  —  7  ax. 

19.  7  mV—  49  m-^x  +  42  m^  and 

14  a^mx^  + 14  a^mx^  —  56  a^mx  —  56  a^m. 

20.  48  s^to*  -  162  ^tx^  +  54  s^^  and 

18  ^fu -9  s  fux - 48  ^fux" -f  24 sH'uo^. 

21.  4a;*-12a^2/  +  5a^2/^-fl2a;i/3-92/*and 

12  a;^  -  36  a:^^^  + 11  a;y  +  48  aJ2/3  _  36  2/4. 

22.  x^-x^y-llxy^-4:f  and  a^^+a^y  - 12  a^/  -  30  a;^  -  8  /. 

23.  The  H.  C.  F.  of  any  number  of  expressions  must  be  a 
factor  of  the  H.  C.  F.  of  any  two  of  these  expressions.  Why  ? 
Must  it  be  the  H.  C.  F.  itself  of  any  two  of  these  expressions  ? 
Explain. 


78-79]  LOWEST  COMMON  MULTIPLE  105 

FindtheH.C.F.  of: 

24.  a^  +  4a3  +  4a2,  a^b-4:ab,  and  a*6  4- 5  a^fe  +  6  a%. 

25.  Sx*-9a^+6a^,   a^-9x^-\-26  x-24:,  and a:^- 8  0^24. 19 a; -12. 

26.  a  +  a^a;  —  2  a^,  a  +  3  a^a;  +  4  ax^  +  2  a;^,  and 

2a3  4-3a2a;4-2aa^-2a^. 

II.   LOWEST  COMMON  MULTIPLE 

79.  Multiples  of  algebraic  expressions.     A  multiple  of  an 

algebraic  expression  is  another  algebraic  expression  that  is 
exactly  divisible  by  the  given  one ;  hence  it  contains  all  the 
prime  factors  of  the  given  expression.  A  common  multiple 
of  two  or  more  algebraic  expressions  is  a  multiple  of  each  of 
these  expressions. 

j5^.^.,  12  a^Q^{]f'—  1)  is  a  common  multiple  of  3  aVQy  +  1) 
and  2a%(^  — 1). 

The  lowest  common  multiple  (L.  C.  M.)  of  two  or  more 
algebraic  expressions  is  the  algebraic  expression  of  lowest 
degree  which  is  exactly  divisible  by  each  of  the  given  expres- 
sions ;  hence  it  contains  all  the  prime  factors  of  each  of  the 
given  expressions,  but  no  superfluous  factors. 

E.g.^  a  common  multiple  of  2  a%'^o[^  and  3  a^a^y^  must  con- 
tain the  factors  2,  3,  a^,  5^,  x^^  and  ?/*  ;  it  may  contain  other 
factors  also,  but  it  need  not  do  so.  Therefore  6  a%^j[^y^  is  the 
lowest  common  multiple  (L.  C.  M.)  of  2  a^h^a:^  and  3  a^j(^y^. 

So,  too,  the  L.C.M'.  of  12  m\x^-lc^)  and  8  JV.2(s+0(^-^)^ 
is  24  m%\s  -\-t')(x  —  k')^(^x  -|-  ^),  —  show  that  this  last  expres- 
sion contains  all  the  necessary^  but  no  superfluous^  factors. 

The  procedure  for  finding  the  L.  C.  M.  of  two  or  more 
expressions  whose  prime  factors  are  known  (or  easily  found) 
may  be  formulated  thus  : 

To  the  L.  C.  M.  of  the  numerical  coefficients  annex  all  the 
different  prime  factors  that  occur  in  the  given  expressions^  and 
give  to  each  of  these  factors  the  highest  exponent  which  that 
factor  has  in  any  of  the  given  expressions. 


106  HIGH  SCHOOL  ALGEBRA  [Ch.  VII 

EXERCISE  LIII 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of : 

1.  8  a'h'',   24  a^6V,   and  18  ahc\  5.    x"  -  ?/'  and  aj^  +  2  ic?/  +  if. 

2.  15a^6^  -20a26V,   and  30  ac^         6.   21  a;^  and  7  ^^^(x -h  1). 

3.  lQ>o?h\  24:aMc,   and  36  a^ft^dl        7.   ic^-l  and  a^  +  a;. 

4.  18a^6r^,  12 pV^,  and— 54a6y.      8.   4:X^y —  y  Siud  2  x^ -\-x. 

9.    Is  12a^6''(a^  — 1/^)  a   common   multiple   oi  2  a^b {x  —  y)  Siud 
3  ab\x  -y)?     Is  it  their  L.  C.  M.  ? 

10.  What  factors  must  an  expression  contain  in  order  that  it 
may  be  a  common  multiple  of  two  or  more  other  expressions  ? 
that  it  may  be  their  L.  CM.? 

11.  Are  both  6  ax^  and  —  6  ax"^  multiples  of  3  a;  ?  Explain. 
If  a  multiple  of  an  expression  has  its  sign  reversed,  does  it 
remain  a  multiple  of  the  given  expression  ? 

12.  Does  a  change  in  the  sign  of  an  expression  affect  the  de- 
gree of  the  expression  ?  If  the  L.  C.  M.  of  several  expressions 
has  its  sign  reversed,  it  may  still  be  regarded  as  their  L.  C.  M. 
Why  ?     (Cf.  Exs.  14-15,  p.  99.) 

FindtheL.  C.  M.  of: 

13.  a  +  &,   a  —  b,   o?  +  W,   and  a*  +  b\ 

14.  3  +  a,   9  -  a^   3  —  a,   and  5  a  + 15. 

15.  a^  —  if,   :ii?  -\-xy  +  2/^,   and  a^  —  xy. 

16.  4a  +  4&,    Q>o?-24.b\   and  a--3  a&  +  2  ftl 

17.  a?  -f  2/^,   ^y  —  y^,   and  a;^  —  2/^. 

18.  y'^-^y  +  Q  and  i/^ - 7 2/  +  10. 

19.  a^— (a+6)a;4-aft  and  a;-— (a— 6)a;  — a&. 

20.  3s2-7s  +  2and6-s-s2. 
Hint.     6  -  s  -  s^  =  -  1  (s2  +  ^  _  6). 

21.  c2-4c4-4,   4-c2,   andc^-16. 

22.  3p«-13p4-14  andl3p-5i)2-6. 

23.  r^"  —  s^"  and  (.s«  —  r^'f. 

24.  (m  +  ny—p^  and  (m  +  w  +i))^. 


79-81  j  LOWEST  COMMON  MULTIPLE  107 

25.  63^262-46-8,  86-12  +  62-63,   and  6^  +  4  6^-3  6-18. 

26.  Find  the  L.  C.  M.  of  each  of  the  sets  of  expressions  in 
Exs.  19-25,  p.  100. 

80.*  The  L.  C.  M.  of  two  algebraic  expressions  found  by  means  of 
their  H.  C.  F.     The  use  of  the  H.  C.  F.  in  finding  the  L.  C.  M.  may 

be  shown  as  follows : 

Let  it  be  required  to  find  the  L.  C.  M.  of  3x*  —  a:r^  —  x'^-\-x  —  2 
and2ar^-3a;2-2a;  +  3. 

By  §  76  it  is  found  that  the  H.  C.  F.  of  these  expressions  is 
x^  —1',  they  may,  therefore,  be  written  thus  : 

Sx*-x^-x^-\-x-2={x''-l)(3x'~x  +  2), 
and  2aj3-3a;2-2aj-f-3  =  (x2-l)(2x-3), 

wlierein  Sx-  —  x-\-2  and  2  x  —  3  have  no  common  factor.  Hence 
the  L.  C.  M.  of  the  given  expressions  is 

(x'-l)(3x^-x  +  2)(2  X  -  3). 

This  shows  that  the  L.  C.  M.  of  the  given  expressions  may  he 
found  by  dividing  their  product  by  their  H.  C.  F. 

Obviously,  the  L.  C.  M.  of  any  other  pair  of  expressions  may 
be  found  in  the  same  way  ;  hence. 

To  find  the  L.  C.  M.  of  two  algebraic  expressions,  divide  either 
of  the  given  expressions  by  their  H.  C.  F.  and  multiply  the  other 
expression  by  the  resulting  quotient. 

81.*   The  L.  C.  M.  of  three  or  more  expressions.     The  L.  C.  M.  of 

three  or  more  algebraic  expressions  whose  factors  are  not  easily 
found,  may  be  obtained  by  first  finding  the  L.  C.  M.  of  two  of  the 
given  expressions,  then  the  L.  C.  M.  of  that  result  and  another  of 
the  given  expressions,  and  so  on. 

EXERCISE  LIV 
Find  the  L.  C.  M.  of : 

1.  a;3_6a^_^lla;_6anda.'3-9a^  +  26a;-24. 

2.  a^-  5x--4i»  +  20anda^  +  2ar^-25a;-50. 

*Articles  80  and  81,  with  Exercise  LIV,  may,  if  the  teacher  prefers,  be 
omitted  till  the  subject  is  reviewed. 

HIGH  SCH.   ALG.  — 8 


108  HIGH  SCHOOL  ALGEBRA  [Ch.  VIl 

3.  2  2/3  -  11  /  -h  18  ?/  -  14  and  2  2/^  +  3  /  -  10  2/  -h  14. 

4.  6a^x-o  a?x  -  18  aa;  -  8  a;  and  6  o?h  -  13  c^h  -  6  a6  +  8  5. 

5.  4  a;^  - 17  ^f  4-  4  /  and  2y^-Qi?y-?,  xY  -^xf-2y^. 

6.  2a;^-9a^  +  18i»2-18a;  +  9and3a;*-llar'H-17a^-12a;+6. 

7.  If  ^,  B,  and  0  stand  for  any  three  given  expressions,  and 
if  i»f  is  the  L.  C.  M.  of  A  and  B,  while  iV^is  the  L.  C.  M.  of  M 
and  (7,  show  that  iVis  the  L.  C.  M.  of  A,  B,  and  (7;  that  is,  show 
that  N  contains  all  the  factors  necessary  in  such  a  multiple,  and 
no  superfluous  factors. 

Find  the  L.  CM.  of: 

a    s*-2s^-i-s^-l,   s'-s^^2s-l,   ands*-3.s2-fl. 
9.   c3  +  3c2-6c-8,    c^-2c--c  +  2,   andc^  +  c-G. 

10.  a52-4a2,  ar5  +  2aa^^-4A-}-8a^  anda^-2aa^  +  4a2a;_8a3. 

11.  a3-j-7a2  +  14a  +  8,  a^  +  Sa'-e  a-S,  and  a3  +  a'-10a+8. 

12.  Ar^-9  A:24.23  k^W,  k'+k'-17  k+W,  and  ]i^+7  k'+7  k-15. 


CHAPTER   VIII 
ALGEBRAIC  FRACTIONS 

82.  Definitions.  An  algebraic  fraction  is  an  indicated  divi- 
sion in  which  the  divisor  is  an  algebraic  expression  :  the 
dividend  may  be  either  an  algebraic  or  a  numerical  expres- 
sion.    (Cf.  §  8.) 

Here,  as  in  arithmetic,  the  fraction  A-i-B  is  usually  written 

A 

in  the  form  —  or  A/B ;  A  and  B  are  called  the  terms  of  the 
B 

fraction,  A  being  the  numerator  and  B  the  denominator. 

If  A  is  exactly  divisible  by  B,  then  A/B  is,  in  reality,  an 

integral  expression,  but  is  written  in  the  form  of  a  fraction. 

E.g.^—- , — -•,and  —  are  algebraic  fractions;  while 

ab—oj^    m—2n        -.a^— a;^  .    ,         . 

, ,  and are  integral  expressions  written 

a  1  a—  X 

in  fractional  form. 

If  both  terms  of  a  fraction  involve  the  same  letter,  and  if 

the  numerator  is  not  of  lower  degree  than  the  denominator 

(in  this  letter),  then  the  fraction  is  said  to  be  improper ; 

otherwise  it  is  proper.     An  expression  that  is  partly  integral 

and  partly  fractional  is  called  a  mixed  expression. 

E.g.^  ^ —  and  — — —   are  improper  fractions,  and 

X  —  1  a 

4  2:  —  3  H — ^^——  is  a  mixed  expression. 
x  —  \ 

83.  Operations  with  fractions.  The  reduction  of  fractions, 
and  the  various  operations  with  fractions  (addition,  subtrac- 
tion, etc.),  are  essentially  the  same  in  algebra  as  in  arith- 
luetic. 

109 


110  BIGH   SCHOOL   ALGEBRA  [Ch.  Vlll 

A  0 

Thus,  if  —  and  ~  are  any  two  fractions  whatever,  then 

^  ^      B   D~BI)'  ^  ^  B^I)~b'~C' 

These  formulas  state  the  rules  for  finding  the  product  and 
quotient,  respectively,  of  two  fractions  ;  the  pupil  may  trans- 
late each  formula  into  verbal  language. 

(i)  The  proof  of  (1)  follows  directly  from  the  definition 
of  a  fraction  (cf.  §§  82,  8). 

Thus,  let  —  =  a;and  —  =?/, 

then  A  =  X'B  •Awdi  0=y'D,  [§§82,8 

hence  A- C=  xByB  =  xy'BD,  [Ax.  3 

AO 
and  therefore  — —  =  xy  [Ax.  4 

BB 

^4    O  fsince   A/B  =  x 

~  B'  B'  L    and  C/B  =  y 

A    C^AO 
B'  D     BB" 
which  was  to  be  proved. 

(ii)  To  prove  (2)  above,  let  -  -  -  =  t, 

B     B 

then  |  =  ^'|'  [§§82,8 

hence  A.^=t.^.R  [Ax.  3 

B    Q         BO  •- 

n 


i.e.^ 


I.e., 


^'{b'o)^*'  ^^^^  ^^^^^ 


and  therefore  A^^=4.^ 

B     B     B    O' 

which  was  to  be  proved. 

Kemark.  The  reciprocal  of  any  given  number  is  1  divided  by 
that  number;  e.g.,  the  reciprocal  of  3  is  ^. 

Hence  it  follows  from  (ii)  that  the  reciprocal  of  a  fraction  is 
that  fraction  inverted. 

Note.  Observe  that  the  validity  of  §  42  is  assumed  in  the  proofs  of  (i) 
and  (ii)  above. 


83-85]  ALGEBRAIC  FRACTIONS  111 

84.   Reducing  an  improper  fraction  to  a  mixed  expression. 

This  change  in  form  is  made  in  algebra   precisely  as  it   is 
made  in  arithmetic. 

E.g.f  just  as  -ijQ=3|^,  i.e.,  3  + J,  so,  too,  since  a  fraction  is  an 

.,.",,...         x^-^-'Jaf  +  B  '       ,-,,     4r-2x 
indicated  division,    — -^ -—  =  a?  +  1  +  — -. 

EXERCISE  LV 

Reduce  each  of  the  following  improper  fractions  to  an  equal 
integral  or  mixed  expression,  and  explain  your  work : 
a^  —  2ab-\-ac 


8. 


2. 


a 

3ar^+9a;  +  2 

3a; 


^     2a;'^  +  4aa;  +  2a^  __ 

x-{-a 


2s-l. 


11. 


12. 


ar^-ar^-2ar^- 

-2a;-l 

a^-x- 

1 

6-hQc-oc^- 

-2c3  +  c* 

c'-3 

8a;''-10a;2_3a;  +  5 

4x^^-3 

3a,6_^2a;-5 

a;»  +  2a;4-l 

15^^-13  i;2- 

8i;-l 

A:  +  2  Sv'-^  +  S'y  +  l 

g^  +  g^  +  l  7fe^-l 

a  +  1  *    Jc'  +  k  +  l 

a^4-7a;2_5  I8a;^-a^-2a;^- 7 

a;2_i       •  ■»•*•  ^3_:3^_^1 

^3 2  a  4- 1 

15.  Is ^^^-^  a  proper  or  an  improper  fraction  ?     Why  ? 

16.  Write  the  reciprccal  of  11 ;  of  —a;  of  —  |;  of  each  frac- 
tion in  Exs.  1-5  (cf.  Remark,  §  83). 

85.  Reducing  fractions  to  lowest  terms.  In  algebra,  as  in 
arithmetic,  a  fraction  is  said  to  be  in  its  lowest  terms  when 
its  numerator  and  denominator  have  no  common  factor. 

Hence,  to  reduce  a  fraction  to  its  lowest  terms,  divide 
both  numerator  and  denominator  by  their  H.  Q.  F.  Instead  of 
dividing  at  once  by  the  H.  C.  F.,  we  may,  of  course,  divide 


112  HIGH   SCHOOL  ALGEBUA  [Ch.  Vlll 

by  any  common  factor,  then  by  another,  etc.,  until  all  com- 
mon factors  are  divided  oat.  Multiplying  or  dividing  both 
terms  of  a  fraction  by  any  given  number  leaves  the  value 
of  the  fraction  unchanged;  for,  whatever  the  algebraic  ex- 
pressions represented  by  A,  B,  and  m, 

^  =  i.^  [§83   (i) 

Bm     B   m  l^         \  J 

=  — ,  [since  m/m  =  1 

which  was  to  be  proved.  -" 

3  ax^     3  ax 

E.g.,  ~ = ,  which  is  in  its  lowest  terms ;  so,  too, 

4  bxy      4  by 

,  ^~^ —  =  (a?  +  l)(a;-l)  ^ x±l^    ^^.gj^  .g  .^  .^g  lowest  terms. 
a^-2a;  +  l      {x-l){x-l)     x-l' 


EXERCISE  LVI 

Beduce  each  of  the  following  fractions  to  its  lowest  terms : 
a^  —  ah  m^  -\-2  mn  -f-  w^        „  m?  -\-if 


a^  —  h^  m®  +  n^  x^  +  x^y"^  +  y^ 

34«^6V  2a^  +  3a;  +  l  ^      3a^-2a-l 

Sla^ft^c*  '     a^  +  5x  +  4  *  '    l  +  a-a^-d"' 

c'-d^  {r-qY-s^  a*-a^-20 

{c-df'  '     (r-q-sf'  '    a'-9a'-{-20' 

10.    May  equal  factors  be  canceled  from  the  numerator  and 
denominator  of  a  fraction  ?     May  equal  parts  (or  factors  of  parts) 

be  thus  canceled  ?     Is  1^L±^  equal  to  -^?     Is  P^^^^, 
obc-^x  obc  ox  —  on^ 

equal  to  ^  +  ^y  ?    Explain  fully. 

Eeduce   the   following  fractions  to  their  lowest  terms,  and 
check  your  work  by  §  25: 

n      ^-t\  T«     xy-zy-x  +  z 

^^'  '^r~t'  ^®-      f^^ 

c3-17c^  +  72c  50-40m  +  8m^ 

c2(cd4-16-2c-8d)*  '        125-8m« 


85-86]                             ALGEBRAIC  FRACTIONS  '  113 

16.    ^^:^ 21.    ^"-^ 


(a -«)(«- 6)  l  +  Ai^"* 

(s-2)(8-3)        .  (a^-5)(a.-  +  2)      , 

(2-s)(3-s)(s-4)  •    aj3-7a^  +  2a;  +  40 

18    p^-llp  +  24  3a^  +  8a-3 

56+i)-i)^  3a3  +  17a2  +  21a-9 

^!±l!.  24      10^  +  20  a:'-a?-2 

*    y'-x^'  '    3a^  +  6a^  +  21a;+42' 

86.  Reducing  fractions  to  equal  fractions  having  given  de- 
nominators. Since  multiplying  both  terms  of  a  fraction 
by  the  same  number  does  not  change  its  value  (§  85),  there- 
fore any  given  fraction  may  be  reduced  to  an  equal  fraction 
whose  denominator  is  any  desired  multiple  of  the  given 
denominator. 

E.g.,  to  reduce  - — -  to  an  equal  fraction  whose  denominator 

shall  be  12  cx^y,  multiply  both  terms  of  the  given  fraction  by 
12 cx^y -7- 4:X^,  i.e.,  by  3  cy, 

EXERCISE  LVII 
Find  the  required  part  in  each  of  the  following  equations  : 
1     ^  =  L.  1     m-2_(m-2y 


3. 


4      12 
Sab^   ? 

4        12^' 
2  cd     16  cH^ 


8. 


6t 

9 

3c- 

2d' 

1           ? 

Scd* 

4:X 

9 

1 

7x^-5 

a-b 

a'-b' 

10. 

5.  ^  =  ,  ;    ^'  11. 


a-\-b  ?  "■    2(2a;-5)      -6x{5-2x) 


9m 

9 

—  r 

9 

r2  +  3      r^  +  10r2+21 

2u-v 

? 

^f +2  V     Su 

^j^5uv-2v'' 

u^  —  uv-\-v^ 

_2{u'  +  if)^ 

lu^-1 

9 

3m-8 

9 

2a;-5      - 

-2.TH-5 

3m-8 

9 

114  '  HIGH  SCHOOL  ALGEBRA  [Ch.  VIII 

13.  If  the  denominator  of  a  fraction  is  multiplied  by  any  given 
expression,  what  must  be  done  to  the  numerator  in  order  to  pre- 
serve the  value  of  the  fraction  ? 

14.  Change     {  to  an  equal  fraction  whose  numerator  is 

J9^  — 9 

— 3j9  — 2;  to  one  whose  numerator  is  3jp^''H-5p^4-2j9;  to  one 
whose  denominator  is  9p— p^;  to  one  whose  denominator  is 
3y4_p2_27p_9. 

87.  Reduction  of  fractions  to  common  denominators.     To 

reduce  any  given  fractions  to  equal  fractions  having  a  com- 
mon denominator  it  is  necessary  only  (1)  to  choose  some 
common  multiple  (§  79)  of  the  denominators  of  the  given 
fractions  as  the  new  denominator,  (2)  to  divide  this  com- 
mon multiple  by  the  denominators  of  the  given  fractions  in 
turn,  and  (3)  to  multiply  both  terms  of  the  given  fractions 
by  the  respective  quotients  (cf.  §  86). 

3  h  hn 

E.g.,  to  reduce    -^^ —  and  — -  to  equal  fractions  having  a  com- 
2  ax  3  x^ 

mon  denominator,  we  choose  6  ax^  as  the  new  denominator,  and 

find,  by  §  86,  that 

3h  _  9  hx        T    hn  _2  dbn 

2  ax     6  ax?  3  x-      6  ax^ 

The  lowest  possible  common  denominator  is,  of  course,  the 

L.  C.  M.  of  the  given  denominators. 

EXERCISE  LVIII 

Reduce  the  following  to  equal  fractions  having  the  lowest  pos- 
sible common  denominator: 

1.  -, — T'  '^"^^  -r—^' 

111     inr  5  m"* 

^     3a  +  l       T  3i»-f4 

2. —  and  ■ 

4  6 

^     9-3  a        ,34-5a^ 

3. and — ^    „   - 

16  h  20  h^ 


4. 

«  +  ^and^-^ 
a  —  h           a-\-h 

5. 

^,  ¥,  - 1^ 

6. 

rfyl|."0|^ 

86-88] 


ALGEBRAIC  FRACTIONS 


115 


7.    -^:z^and         ^  +  ^ 


ar  +  aic  2  aa^  —  2  a^a; 

and 


9.    — !-^^  and  '  -^ 


10. 


a;2  +  a-2/  +  t/'^ 
and 


11. 

12. 
13. 
14. 
15. 

16. 


(m  — l)(m  — 2)  m  — 3         *    x  +  y'x  —  y^  x^  —  xi/ 

Hint.    First  multiply  both  terms 
of by  —  1,  so  as  to  arrange 


and 


2  —  m'    m  +  2'  m-  — 4 


2  -  m 
the  denominators  in  the  same  order. 


m 


and   -^ 


m  —  n     rr  —  m' 


m  4-^*' 


6x 


2  J       3 

.  and 


ar^-1 
3  a 


and 


3  a  — 6  X 


^  ^-2  and       ^  +  ^       , 

5  —  a'  a^  —  8  a  +  15'  a^  —  6  o  +  5 


and 


(aj-l)(a;-3)  (x-8)(3-a^) 

17.    Show  that 


Hint. 


equal 


(x-S)iS-x) 

-7 


(x-S)(x-S) 
Cf.  Hint,  Ex.  11. 


(c-l)(2-c)      (c-l)(c-2) 
2-a;  x-2 


18.    Show  that  _        ^^        ox  —  /        Kx/       ox- 

(5  — x-)(aj  — 3)      (a;-5)(a;— 3) 

Reduce  to  equal  fractions  with  the  lowest  common  denominator ; 

? , ? ,  and — (cf.Ex.l6). 

(a;  _  7) (a?  -  2)'  (2  -  x)(x  -  4)'         (4  -  aj)(a;  -  7)  ^  ^ 

1  2.-3 


19. 
20. 


and 


(.y  _  2)(v  -  5) '  (v  -  5)(3  -  ?;)'         (v  -  3)(7  -  v) ' 


21. 


a+  5 


a-2 


and 


a  +  1 


a2_  4  a +  3'  8  a- a2_  15'  6a -5- a^ 


88.   Addition  and  subtraction  of  fractions.     From  §  38  it 
follows  that 

a      h      a  +  h        -.a      h      a  —  h 
-  -4-  -  = and = ; 


c     c 


c      c 


116  HIGH   SCHOOL  ALGEBBA  [Ch.  VIIl 

that  is,  ill  algebra,  as  in  arithmetic,  the  sum  (or  difference^  of 
two  given  fractions  which  have  a  common  denominator  is  a 
fraction  whose  numerator  is  the  sum  (or  difference^  of  the  given 
numerators,  and  whose  denominator  is  the  common  denominator 
of  the  given  fractions. 

^       m^        2h  ^m^^2h        .       2  a b^c      ^2a-l?c 

'^'"^ax     Sax        Sax    '  ^^    6(x-l)      5(a^-l)      5(a:-l)* 

If  the  given  fractions  have  unlike  denominators,  they  must, 

of  course,  be  reduced  to  equal  fractions  having  a  common 

denominator  (§  87)  before  they  can  be  added  or  subtracted. 

3  7 

Ex.  1.     Find  the  sum  of  and 


x  —  2  x-\-l 

Solution.     The  L.  C.  M.  of  the  denominators  is  (x  —  2)  (a;  -j- 1) 


and,  by  §  87, 


and 


3     ^       3(a;+l)       ^        3a;  +  3 
x-2      (x-~2)(x  +  l)      (aj-2)(a!  +  iy 

7     ^       7(a;-2)       ^       7a;-14 
x  +  1      (x  +  l)(x-2)      (a;-fl)(a;-2)' 
7         Sx  +  3-{-7x-U         10a!-ll 


+ 


x-2     aj  +  1         (x-{-l){x^2)        (a;  +  l)(a;-2) 

7  3 

Ex.  2.     Subtract from 


x-^1  x  —  2 

Solution.     Proceeding  as  in  Ex.  1,  we  obtain 

_3 7     ^3a;  +  3- (7a;-14)^     -4a;  +  17 

x-2     x-\-l  (^x-\-l)(x-2)  (^x  +  l)(x-2) 

Note.  The  minus  sign  before  the  second  fraction  means,  of  course,  that  all 
of  this  fraction  is  to  be  subtracted,  hence  the  need  of  the  parenthesis  in  the 
numerator  of  the  next  fraction. 

EXERCISE  LIX 

Simplify  the  following  expressions  and  check  your  results : 

3    ^4.^  5    a?-l      x-^3     x-\-7 

■    3  "^6  •             '      2            5    "^   10   * 

.     a  +  S     a-{-5  c-^-d b_ 

5  7     *                              '       d        2d' 


88]  ALGEBRAIC  FRACTIONS  117 

-        1      +^.  16.  1  3 


x+y     x  —  y  2s^—s  —  l      6  s-  — s  — 2 

-        X             a;      #  _  _     ct^  —  aa.'  H-  0^      a  +  a; 

1  —  a^     1  +  a^  a^  +  ax  +  a;^     a  —  x 

10.  ^Lz^H-^^I^  +  ^r:i^.  [Suggestion.     x  =  ^.] 

ah          he          ac  L                             1  J 

11.  r-j_^r+^__r^--^>  ^^    _a c^ ^  s. 


2rs  s  —  as^  —  w 


(a5-2/)^     a;^  +  4a;?/-5/  a^-7a;  +  12     a^-5a;+6 

13.  ft  +  ^-c  a-6  +  c^    21.       1      I       ^-?/  a:^-a;y^ 
a?  —  ih  —  cf     {a—hy—c^           '    x-\-y     x^—xy-\-y^     3^-\-'if 

14.  J~-^ 22.    -i-  +  -^ 1. 

a;2_i      aj2_^_2  s(s  —  t)      t{s  +  t)      st 

15  a?  4- 7 a;  +  2  ^^    2a;-3c     2a;-c     ^^ 

a^_3a;-10     a^4-2a;-35'       *     a;-2c        x-c 

Remark.  Since  a  fraction  is  a  quotient,  its  sign  (the  sign 
before  the  fraction)  is  governed  by  the  law  of  signs  in 
division;  hence,  whatever  the  expressions  represented  by  a 
and  hy 

__a_  —  g  _   a 
h~    b    ~-h' 
Exercises  in  subtraction  may  therefore  be  changed  into  exercises 
in  addition ;  the  results  of  such  exercises  are  often  called  algebraic 
sums  (cf.  §  16). 

E.g.,  Ex.  23  may  be  written  ^^~^^  +  ~^^"^^  +  3 a;. 

X— 2 c         x~c 

24.  Write  Exs.  6,  8,  11,  13,  14,  15,  19-22,  above,  as  exercises 
in  addition. 


*  Cf .  Ex.  2,  Note. 


118  HIGH  SCHOOL  ALGEBRA  [Ch.  VIH 

Write  the  following  as  exercises  in  addition,  and  find  the  alge- 
braic sum  in  each  case : 


a      a  + 1      a  +  2                          s—1      2(s  + 1) 
26. h  «  —  :i 30. 1 


ic  —  1  1  —  X  a  —  1  a(a  —  1) 

27.    -i L.-.!^.        31.  1  1 


a  +  6  a-6      5'^  -  a^  2a^-a;-l      3-aj-2a^ 

d  cd  cc?2  ^^    2h-a      3x(a-b)  ,  ?^-2a 

G  +  d  {c-\-df     (c  +  df  x-b         b'^-x"         x  +  b 

Simplify : 


{a-b){a-c)      {b-c){b-a)      (c-a){c-b) 
[Hint.     The  given  expression,  witli  the  letters  in  alphabetical  order,  is 

^i + -I + J .1 

(a-6)(a-c)      (,b-c)(a-b)      (o-c)(6-c)-l 
34.    "" + * +  <' 


(a  —  b)(a  —  c)      (b  —  a)(b  —  c)      {c  —  a){c  —  b) 

35  a;-l  2(a^-2) x-S 

(x-2)(x-3)      {3-x){x-l)      (x-l)(2-x) 

36.  .,   .  ^  ..  „-  „    .  V  „  „+        ' 


x^  —  5  xy  -\- 6  y^      x^  —  A  xy -{- 3  y-     x'  —  3xy-^2y' 
37.    ~ H f-  ^ 


x^  —  5x-{-6      3a7  —  2  —  ic-      4a;  —  3  —  a;'-' 
38.  ^^  [  ^^  I  «^ 


(a-c)(a-6)      (6_c)(6-a)      (c-a)(c-6) 

89.   Reducing    mixed    expressions    to    improper    fractions. 

Since  an  integral  expression  may  be  written  in  the  frac- 
tional form  with  the  denominator  1,  therefore  reducing  mixed 
expressions  to  improper  fractions  is  merely  a  special  case  of 
addition. 

E.g.,x  +  1+     1     ~»'  +  l  .      1 


a;  — 1         1  a;  — 1 

x—1  x  —  1     x  —  1 


88-90]  ALGEBRAIC  FRACTIONS  119 

EXERCISE    LX 

By  the  method  of  §  89  simplify  the  following  expressions  : 

af  —  l  a  +  2b 

,  -i        2x  „  9        oX'^-j-oi^  — x-\-l 

2.  x-\-l •  7.  X  — or  —  XT ■ — 

x  —  1  l-\-x-{-x~ 

c^— 2c  +  l  1  —  2a;  +  a;- 

a4-&  2aH-36 

_    -,               2      1— V^                 -.r*    -1                I,        ax-\-bx-\-ab 
5.   1  —  y  —  if  — ^  •  10.    1  —  aa;  —  6ir :; — 

1  —  2/  1  —  ax 

11.  May  the  numerator  in  the  answer  to  Ex.  1  above  be  found 
by  multiplying  x~l  by  a^  —  1  and  adding  x'^  to  the  product  ? 
Explain  this  method  fully  (of.  §  84,  also  Ex.  20,  p.  50). 

12.  By  the  method  of  Ex.  11,  solve  Exs.  2-5,  and  8-10,  above. 

90.  Product  of  two  or  more  fractions.  In  §  83  it  was 
shown  that,  whatever  the  expressions  represented  by  A^  B, 
(7,  and  i>, 

A     O^AO 
B  '  J)     BD' 
This  principle  is  easily  extended  to  finding  the  product  of 
any  number  of  fractions  ; 

A   ^  ^  ^  =  A^  ^   Oi^ACE   a^ACEa 

^*^*'    B'  d'  f'  H     BB'  f'  H     BBF  '  H     BBFH 

Hence,  the  product  of  two  or  more  fractions  is  a  fraction  ivhose 
numerator  is  the  product  of  the  numerators  of  the  given  fractions^ 
and  whose  denominator  is  the  product  of  their  denominators, 

Ex.  1.     Find  the  product  of  ^^^  ~    ^  and      ,  ^^  ^^  • 

'6xy  a  (or  —  1) 


SOLUTION 

a(x  —  l)  6x       _  Gax{x  —  \)    _ 


3xy  a(ar  — 1)      Saxy{x^  —  1)      y{x  +  l) 


[§85 


120  HIGH  aCHOOL   ALGEBRA  [Ch.  VllI 

Remark.  Observe  that  the  factors  3,  a,  x,  and  a^  —  1  might 
have  been  "  canceled "  even  before  the  multiplication  was  actu- 
ally performed.  Pupils  should  cancel  wherever  possible,  and  thus 
simplify  their  work. 

EXERCISE  LXI 

Find  the  following  products,  and  simplify  your  results : 

2.  ^  .  ^.  9. 


xyz     ac^  m-  —  mn      ef—f^ 

g    Qxy     l^yh\  ^^  x-1        .a;  +  l. 

*    82    *     9a^    *  '  a^  +  2a;  +  l     x-1 

^    -  5  iH^      Sr^t  ^^  x-1            x2  +  4a;  +  3 


12iu^      lOs^v^  3aj2-|.8i»  +  5          ar^-1 

—  Ix^yz     f      6  fey^Y  /-^  —  ?-8  +  s^         j?  —  <f 

IS  xyz'-    '\     11  x'zj  '      {p-qf     *r^  +  ?V  +  «^* 

4myi^     —am  (a  —  xY      x^-\-xy  +  y^ 

1          m^n^  x^  —  y^      d^  —  2  ax-\-x^ 

7.    Isx^y^  .  t  .  4-.  14.   i—^r-r  .  a±b+l, 

y     9xY  (a  +  6)2_l     a-b-1 

^m+2  *    ^m    •52*  *      2s^     *     si  +  i^      's2_g^* 

16  ^  +  5     a4-3     gg-Ta  +  lO. 
a-2  *  a-5  '  2a-  +  5a-3* 

17  a.'^  +  y'     a*  +  a'b'  +  b*  x^-f 

'  a^-^b^'        a^-b^       'x*-2afy''-\-y*' 

1^  _  f       3 
18-  -^ — r  •  r^r-:  =  ?    May  i^  be  canceled  in  this  example  ?     Ex- 

plain  (cf.  Ex.  10,  p.  112). 

19.  Simplify  [x-\-2y )  •  — ^  by  finding  the  product  of  each 

V  y)     a-\-x 

term  of  the  multiplicand  by  the  multiplier,  and  then  adding  the 
partial  products  (cf.  §  32). 

20.  Simplify  [  a;  +  2  ?/  —  -  )  •  — ^  by  first  reducing  the  multipli- 

V  •       2/7     a  +  a; 

cand  to  an  improjjer  fraction. 


9(V01] 


ALGEBRAIC  FRACTIONS 


121 


Simplify,  and  check  your  results : 
22.   (5s2-36s+7) 


25. 


26. 


27. 


28. 


29. 


0  + 


(f  -  ?> 


1-6- 

3s-21* 


23. 


24. 


a  +  25 
2b-a 

1      ^'" 

V 


x'  +  fj      x'-f 


Sa(a-b) 


V       af-9x-^20j\'jf-6x  +  9  J\        x    J 


b(x^ 


a 


9x-^ 
-hx  +  y 


3  a'  +  3  a'bx  +  3  a*by 
ax  —  3by—Sbx-\-ay 

1     ^a  +  b^(a-\-byi\         (a-\-by\a 

a_b__c 2\A  2c      V     5a 

6c      ac      a6      «  A        a  +  6  +  cy  l^  —  a  —  b 


(a  +  6)2fa  +  6  +  c 


30.   What  doesr-Y  mean  [cf.  §  9  (ii)!?     Show  that  (^-Y 

Ve/  \/-i  \o/ 


and  that  ("^  =  i^?!^'  =  ^^ . 


fey     f  rs'  Y     /m  -  2  nV     /-ScV     /a +  6 


31.  Raise  the  following  fractions  to  the  powers  indicated : 

b-c\\    r  2.<i/V  Y 
/  '    \—'Smyip) 

32.  Write  the  following  fractions  as  powers  of  other  fractions : 

6^  +  26  +  1.    _t_.      s^-^sH-^-^sf-f 
a?  '    16aj8'    64  +  48i/  +  122/'  +  2/'* 

91.  Division  of  fractions.  In  algebra,  as  in  arithmetic,  to 
divide  by  any  fraction  gives  the  same  result  as  to  multiply  hy 
the  reciprocal  of  that  fraction  [of.  83  (ii)]. 


a^x 


E.g.,  -^1^  - 


a'^x     by 


by-     by     b-y^ 


ex 


af_ 

bey 


Note.  If  the  divisor  is  an  integral  (or  mixed)  expression,  it  should  be 
written  in  fractional  form  (that  is,  as  an  improper  fraction)  before  proceed- 
ing as  above. 


122  HIGH    SCHOOL   ALGEBRA  [Ch.  VllI 

EXERCISE  LXII 

Simplify  the  following : 

Ud'b'      2a'b'  /^H-8^  +  7       ^  ^ 

a" -121  .  a +  11  ,^     /    _^^^^V  «'       i\  .  '^ 


2-   ^ — r  -^  — ^  •  11.  «  +       /i  .>  '>    -  i  -    2 

•      ^V     '   -5q^'  '  (a+6)2-9  '  a  +  6  +  3 

4.    l^!i£±fU-^.       ,  13.  ^l±^^(t^-^2tv  +  2v^y 

7'.s  —  .9^         r  —  s  —or 

^     oc^  —  a"      (x  —  ay  ,  ^  -y^  —  25        5  — 'i? 

5^ J.-  N z_  ,  2,4. r-  • 

a^  -j-a^       x^  —  a^  v^-\-v-\-l      v^—1 

^       Ux^-7x        2aj-l  „  x^-1          x''-12x+?>n 

15. 


12x^-\-24:X^     x'^  +  2x  lO-^'Sx-a^      x^  +  3x  + 

„    p(^    0  Q„4     5  r)V                          ^     /h         2cd  \      d  —  c 
7.    OO^rrs^-? '- —  16.      1 '  • 


a.  (.^-3.-10).-^;.      1.  ^14-.-^^ 

a^-b*        ,  (g  -  by  r'-l    ,  1  -  r^ 

•    a^  +  a2^2_^6*  •   a'-b'  '  t^-dr  '   3-r' 

0^^-5^-36     V  ^'-y 

20.    2r^-21r-ll^/^_r-3 
^^-^-ISr-T    •  V        r-7 


^. 


22. 


i»2_49  a;2_^_42 

5  m^n  —  5  ri^ 


urn  +  2  mrr  +  n^      V  wi 


23    2a^  +  13.T  +  15  .  2a?^  +  lla;  +  5 

4x^-9  ■         4x2-1 

2^      x^-nx-  +  l^    .  /       a^2_3^_4 


9rt^-34a2  +  2o      V      3a2  +  8a  +  5 


91-92]  ALGEBUAIC  FRACTIONS  123 


{p  —  qY       P  —  Q       p^-\-q^  p^  —  q'^ 


25-  f:^=^,^^~^^  +  z^  ■  P-'P\^1  (cf.  §  10). 


I'oTj—WjfY  ^  (^  —  ^y^     P^^  —  4  py—^p^y+px 


' \p-q  J 


p^  —  pqj       \p—qj  ^ -\- ^  xy -\- ^  if 

92.  Complex  fractions.  In  algebra,  as  in  arithmetic,  a  frac- 
tion whose  numerator  or  denominator,  or  both,  are  them- 
selves fractional  expressions,  is  called  a  complex  fraction. 

1  1 

a  X 

^'Q-->  ^ and  r- are  complex  fractions. 

X 

Since  a  complex  fraction  is  merely  an  indicated  quotient, 
it  may  be  simplified  by  means  of  §§89  and  91. 

^g    ^     _  ^  _aP  —  1  X  _^  — 1 

'.,.  .1" x^-\-2x-\-l~      X      '  x^-\-2x-{-l~ x  +  1 

X  X 

In  many  cases,  however,  a  complex  fraction  is  most  easily 
simplified  by  first  multiplying  both  its  terms  by  the  L.  C.  M. 
of  their  own  denominators. 

Thus,  in  the  above  example,  multiplying  both  terms  by  x 

gives    o^  ^ T")  which  (by  §  85)  equals     ~    ,  as  before. 

X^  -}-  ij  X  -\~  J~  X  -J-  JL 

EXERCISE  LXIII 

Simplify  the  following  expressions  : 

^  (m  —  nf  -4-i 

s—p  _m-  g.     s      s 

1       =— .  3.    .  9.      • 

■    s  +p  m^  —  n^  1  _  I 

p^  mn  V      s 

c-^d  (x-S)(x-5)  1_1 

^     c—  d  x  —  7  6      ^      -^ 


(^-d^  x-n  i^_j. 

c^-fd^  (a!-3)(.T-7)  e"     f 

HIGH  SCH.  ALG.  — 9 


124 

7. 
8. 

c 

HIGH  SCHOOL  AH 

1  +  7 

9                ^ 

a      6      c 

9EBRA                      [Ch.  VIII 

3      a-c 
11.             «-f 

.      ah' 

x  —  6 
x-2    ' 

X 

a 

(a  -  6)^ 
2a-3&+c 

^         ,.4 

oca 
a 

1+-  +  J 

a  — 
3 

18.      "^^ 

6     a-\-h                  ^^ 
1      ^ 

a^-b'     a^  +  W 
a-{-h      a—h 

15 

&  '  a  +  6 

a  —  b      a  +  b 

1 

1  I      ^ 

a     «-l 

^  +  1  +  0. 
a; 

a+1 

"  1  /  1            ^ 

*  To  simplify  such  an  expression  we  begin  at  the  end  and  work  backward. 
The  first  step  here  is  to  add  ^  to  1 ,  then  divide  4  by  this  sum,  then  add  this 
quotient  to  1,  and  finally  divide  3  by  this  sum,  obtaining  the  result  j\. 


CHAPTER   IX 
SIMPLE   EQUATIONS 

93  Introductory  remarks  and  definitions.  As  we  have 
already  seen  in  Chapter  V,  an  algebraic  problem  states  a  re- 
lation between  numbers  whose  values  are  known  (called 
known  numbers),  and  others  whose  values  are  at  first  un- 
known (called  unknown  numbers).  It  is  by  means  of  this 
relation,  translated  into  an  equation,  that  we  can  find  the 
values  of  the  unknown  numbers. 

Besides  the  numerals  1,  2,  3,  •••  the  letters  a,  5,  <?,  •••are 
often  used  to  represent  known  numbers ;  unknown  numbers 
are  usually  (though  not  necessarily)  represented  by  the  later 
letters  of  the  alphabet,  such  as  ic,  ^,  and  z. 

A  literal  equation  is  one  in  which  one  or  more  of  the 
known  numbers  are  represented  by  letters;  while  in  a  nu- 
merical equation  all  known  numbers  are  represented  by  the 
numerals  1,  2,  3,  etc.  An  integral  equation  is  one  whose 
members  are  integral  in  the  unknown  numbers  (cf.  §  34); 
known  numbers  may  appear  as  divisors  and  the  equation 
still  be  integral. 

By  the  degree  of  an  integral  algebraic  equation  is  meant 
the  highest  number  of  unknown  factors  which  it  contains  in 
any  one  term. 

E.g.,  of  the  equations  (1)  4a;-5y=:10,  (2)  —-8  =  -,  (3)  4ta'^  =  2+x, 

a  h 

(4)  3 x2  -  9  a3  =  4  y2^  and  (5)  |-  -3 a;y  =  5 y,  all  are  integral,  (1),  (2),  and  (3) 

are  of  the  first  degree,  (4)  and  (5)  are  of  the  second  degree,  (2),  (3),  and 

(4)  are  literal,  and  (1)  and  (5)  are  numerical. 

Equations  of  the  first  degree  are  usually  called  simple 
equations,  and  often  also  linear  equations  (cf.  §  140,  Note)  ; 

125 


126  HIGH  SCHOOL   ALGEBRA  [Ch.  IX 

while  equations  of  the  second  and  third  degrees  are  called 
quadratic  and  cubic  equations,  respectively. 

94.  Equations  having  fractional  coefficients.  Equations 
having  fractional  coefficients  may  be  solved  as  follows: 

Ex.  1.     Given  -— —  ^  =  « 5  *^  ^^^  ^• 

Solution.  On  multiplying  both  members  of  this  equation  by 
6  (Ax.  3),  it  becomes        5  oj  —  48  =  3  a;, 

whence  2  a;  =  48,  [Ax.  1 

and  therefore  a;  =  24.  [Ax.  4 

On  substitution  in  the  given  equation,  24  is  found  to  check;  it 
is  therefore  a  root  of  that  equation. 

Multiplying  both  members  of  an  equation  by  a  common 
multiple  of  its  denominators  is  usually  spoken  of  as  clearing 
the  equation  of  fractions. 


EXERCISE  LXIV 
Solve  the  following  equations,  checking  the  root  in  each  case : 

_     2x  —  4     3a;  —  7      o 

O. — =  Zi. 


2. 

X           4:X            f, 

2==Y"^' 

3. 

2a;      5a;      „ 
3        4   ~ 

4. 

3a;  =  |  +  25. 

5. 

7a;     5      w     2a; 
10      6            15" 

6. 

4.      ._23. 
9       6     36 

7. 

7?i  4-  8      m  -f  4     ^ 
6            11     ~   • 

[Cf.  §88,  Ex.  2,  Note.] 

5 

7 

9. 

V+s= 

:91 

-lOv. 

10. 

T-'i- 

=  - 

-i  +  2ia, 

,,     X     2x  ,  2x-3  ^ 

12.   l„-~+-^ — . 

,,    a;  — 3  ,  7x     4x  —  S 
3         18  5 

14.    Are  the  above  equations  integral  or  fractional  ?     Why  ? 


93-95]  SIMPLE  EQUATIONS  127 

15.  How  is  an  equation  cleared  of  fractions  ?  Upon  what 
axiom  does  the  process  depend  ? 

16.  In  each  of  Exs.  5-7  name  three  factors,  any  one  of  which 
might  be  used  to  clear  the  equation  of  fractions.  How  may  we 
in  each  case  find  the  least  factor  which  can  be  used  ? 

17.  Name  the  degree  of  each  equation  in  Exs.  22-28  below. 
Which  of  these  equations  are  simple  ?    quadratic  ?    cubic  ? 

Solve  the  following  equations,  and  check  as  the  teacher  directs: 

^3    7,a;-4^^^     4.x  +  7^  25.   a:^-x=6.     (Cf .  §  72.) 

5  "  7      * 

.      /o        ox  X       26.   x'-15x'  +  mx  =  0. 

19.    -2 a; +  4 -(3 a; 4- 2)=--. 

20    Mzz^  =  ^±2_3^-2  27.    v'-^2v'  =  v  +  2. 

5  G  7      ' 

^,     Ax     21X-25     4.(3x-2)  28.   ^±i ^ _A_  =  ^"zi^S^ 

21.  -3-  =  — 12 2~  ^         -^-^         1^ 

22.  3s-15_2g-^^3         -^^       29. 

8  4  2 


23. 


1.25+ .5  a  ^.25  g- 2.375 
.25  1.125 


7 

'  - 

k-i 

5 

;22- 

-2^- 

15 

:2|  = 

^-3 

3 

2a: 
3 

+  4 

161 

:  — 

—  X 

a-32     2a-3_15-3a  _3 _lb^-x     x 

2  4      ~       8       '      ^^-        2  32' 

31.    J[c2-  i(c2_3)  +  6c-7]  =  9fi. 

-  (i-0(i-^^)-(i-'^)(3-^^)=-^- 

95.  Equivalent  equations,  (i)  Two  equations  are  said  to 
be  equivalent  if  every  root  of  either  is  a  root  of  the  other 
also.  Thus,  the  several  equations  in  the  solution  of  Ex.  1, 
§  94,  are  equivalent  ;  each  has  the  root  24,  and  that  only. 

The  method  of  solution  used  in  Ex.  1,  §  94,  consists  (1)  in 
deducing  from  the  given  equation,  by  means  of  the  axioms, 
a  succession  of  new  and  simpler  equations,  and  (2)  in  finding 
the  root  of  the  last  and  simplest  of  tliem  all. 


128  Jliail  SCHOOL   A  LG Eli  HA  [Ch.  IX 

That  the  root  of  this  last  equation  (24,  in  this  case) 
happens  to  be  a  root  of  the  given  equation  also  is  due  to  the 
fact  that  applying  the  axioms  to  equations  usually  produces 
equivalent  equations. 

(ii)  Although  the  axioms  are  correct,  their  application  to 
equations  does  not  always  result  in  equivalent  equations. 

E.g.,  given  the  equation  3  a;  —  4  =  1. 

Multiplying  both  members  (Ax.  3)  by  x  —  2,  we  obtain 

3a---10aj+    8  =  a;-2. 

Simplifying,  3a^- 11  a; +  10  =  0,  » 

i.e.,  (a;-2)(3aj-5)  =  0; 

whence  (§  72),  aj  =  2  or  a;  =  f ; 

but  2  is  not  a  root  of  the  given  equation. 

The  axioms  must,  tlierefore,  be  used  with  caution,  and 
results  should  always  be  checked. 

(iii)  The  following  changes  in  equations  will,  however, 
always  produce  equivalent  equations  (cf.  El.  Alg.  p.  143). 

(1)  Transposing  and  uniting  terms  (Axioms  1  and  2). 

(2)  Multiplying  and  dividing  by  any  expression  whicli  is 
not  zero,  and  which  does  not  contain  the  unknown  number 
(Axioms  3  and  4). 

96.  Literal  equations.  Literal  equations  in  one  unknown 
number,  and  of  the  first  degree,  may  evidently  be  solved  by 
the  method  already  employed  for  numerical  equations. 

E.g.,  given  the  equation — —  = 3;  to  find  x. 

hah 

On  multiplying  through  by  ah,  to  clear  of  fractions,  the  given 
equation  becomes 

ax-hx-2W  =  a^-Zah.  [Ax.  3 

Hence  ax  —  hx  =  a-  —  'S  ah  -\-2  h^,  [Ax.  1 

i.e.,  {a  —  h)x  =  or  —  3  a6  +  2  ft^j 

and,  therefore,  x  =  ^'-3a6  +  2 h^  r^^  ^ 

a  —  b 
=  a-26. 


96-96]  SIMPLE  EQUATIONS  129 

Check.     On  substituting  a  — 2  b  for  x  in  the  given  equation, 
we  obtain 

a  — 2b     a  —  2b-\-2b^a     g 
b  a  b       ' 

in  which  the  first  member  readily  simplifies,  and  becomes  -  —  3; 
hence  a  —  2  6  is  a  root  of  the  given  equation. 


EXERCISE  LXV 

Solve,  and  check  as  the  teacher  directs : 
1.3cx  —  cl  =  2d.  11    i_l_i 

2.  (a-b)x  =  3b-Sa.  *  e     / 

3.  2  ax  =  0?  —  ex.  12_  _1 i_  _-  J 

4.  «6-  +  c.s  =  a2-c2.  ^-/      ^+/ 

c         7  ^ 


a 


^  -|^  14.  (6  — c)aj  — (a4-&)a;  =  c?  — 20. 

^'  d^^~   ■^^d'  15.  ^^^lA^4.^  =  16  +  i^. 
7.  z-Saz=(l-Say.  b  a  a 

a  cx-c'  +  id'  =  2dx.  16-  3d(aj  +  3cd)=c(c2-a^). 


9. 


cz  +  d     z  +  9cd  17.  ^-2a&_-^^^-3c 


3(^ 


ab 


10.  n^a;-a;  =  n-a;-e.  la  ^nl^  +  ^  ^  ^^!±iA\ 

2  6         a  a6 

19.  Solve  Ex.  1  for  c  ;  for  d.     Similarly,  solve  Exs.  2,  5,  9,  and 
11  for  each  letter  in  turn,  and  check  your  results. 

Solve  the  following  equations  : 

20.  b(c  —  x)-\-a(b  —  x)  —  b(b  —  x)  =  0. 

21.  ah  +  bh  =  3ab-3  s{a'b  +  ab""). 

22.  (a  —b)(x  —  c)  —  (b  —  c)(x  —  a)  =  (c  —  a)(a;  —  &), 

^^.  — 1 1 =  7H 1 r-i- 

b  a  c         b      c     a 

24.  What  is  a  literal  equation  ?     a  numerical  equation  ?     To 
which  class  does  2  u;  —  13  +  ax  =  14  a?  belong  ? 


130  HIGH  SCHOOL  ALGEBRA  [Ch.  IX 

25.  Each  member  of  the  equation  m-  —  5  m  —  24  =  —  3  (m  —  8) 
is  divided  by  m  —  8  ;  are  the  quotients  equal  ?  Why  ?  Show- 
that  the  new  equation  is  not  equivalent  to  the  given  equation. 

97.  Fractional  equations.  Equations  containing  expres- 
sions that  are  fractional  in  the  unknown  number  (§  34)  are 
called  fractional  equations.  The  methods  already  employed 
apply  to  such  equations  also. 

3     15       1 

Ex.  1.  Given  the  equation  -  —  -  = [■-'•>  to  find  the  value  of  x. 

X     2     3x     6 

Solution.  Clearing  of  fractions  by  multiplying  each  member 
by  6  Xj  the  L.  C.  M.  of  the  denominators,  we  obtain 

18-3x  =  10  +  », 
whence  x=2; 

moreover,  this  value  of  x  checks,  hence  2  is  a  root  of  the  given 
equation. 

Ex.2.   Given =  — ? — ;  to  find  a;. 

2(a;-l)      7(x  +  l)      x  +  l      7(0^2-1)' 

Solution.   On  multiplying  each  member  by  2  •  7  (a;-|-l)  •  {x—1), 

to  clear  the  equation  of  fractions,  we  obtain 

3-7(a;  +  l)-2(a;-l)  =  8-2.7  (x-l)-20, 

le.,  21a;  +  21-2a;  +  2  =  112a;-112-20, 

whence  ^  =  |> 

which  checks,  and  is,  therefore,  a  root  of  the  given  equation. 

7     af  —  1 

Ex.  3.  Given  -  +  — =  x:   to  find  x. 

6     ar  — 1 

Solution.  On  multiplying  this  equation  by  6  (a^  —  1),  to  clear 
of  fractions,  we  obtain 

7 (x""  -1)+  6  (x^  -1)  =  6  x(a^  -1), 
i.e.,  7x^-7-\-6x^-6  =  6a^-6x, 

whence  7  a^  + 6a;-13  =0, 

i.e.,  (a;-l)(7a;H-13)  =  0, 

and  the  roots  of  this  equation  are  1  and  —  J/.  [§  72 

On  substituting  these  values  of  x  in  the  given  equation,  it  is 
found  that  —  -y-  checks,  but  that  1  does  not  check ;  hence  —  4^  is 
(and  1  is  not)  a  root  of  the  given  equation. 


96-97]  SIMPLE  EQUATIONS  131 

Note.     This  shows  that  clearing  an  equation  of  fractions  may  introduce 
extraneous  roots,  i.e.,  roots  which  do  not  belong  to  the  given  equation. 

~3  _  1 

In  this  example  the  fraction  might  have  been  reduced  to  its  lowest 

x'^  —  1 

terms  before  clearing  the  equation  of  fractions.     In  that  case  the  multiplier 

6(x  +  1),  instead  of  6(x^  —  1),  would  have  sufficed  to  clear  of   fractions; 

the  unnecessary  factor  x  —  1  brought  in  the  extraneous  root  1. 

No  extraneous  roots  are  brought  into  a  fractional  equation  unless  an 

unnecessary  factor  is  used  in  clearing  of  fractions  (cf.  El.  Alg.  §  99).     Such 

roots,  if  introduced,  are  always  discovered  in  checking. 


EXERCISE  LXVI 

Solve  and  check  the  roots : 

4     0^-3      a;-f  >^^a;  +  26  ^     _3 ?_  =0 

73  6      '  '.T  +  1      x-l~   ' 

ic-l,ir-2     11-13a;  o^/-l      i      1 

23  12  y  +  1  y 

X      16  8a;  10     ^y     by 

10.  ^(2-a;)-|(3-2a:)  =  ^±i^. 

11.  -M_H--^_=-A_  +  3. 
x^  —1      x—1      x-\-l 

12.  Clearing  Ex.  11  of  fractions,  we  obtain  a;^  — 2a;— 3  =  0; 
are  the  roots  of  this  equation,  viz.,  3  and  —  1,  roots  of  the  given 
equation  also  (cf.  Ex.  3,  Note)  ? 

13.  Define  a  fractional  equation.  Which  of  the  equations  in 
Exs.  4-11  are  fractional  ?     Explain. 

Solve  the  following,  and  check  as  the  teacher  directs: 
10a;4-17      5a;-2      12a;-l 


14. 


18  9  11  a; -8 


Hint.     Multiply  both  members  by  18,  and  combine  similar  terms ;  then 
multiply  both  members  of  the  resulting  equation  by  11  aj  —  18. 

5(^-4-4)      7  a;- 3^3  (3 a; +  1) 
11  3a;H-2  22 


132  HIGH  SCHOOL  ALGEBRA  [Ch.  I.\ 

3-2/     8^  2/  +  3       8(2/  +  3) 

17    g-5     g-lQ^g-4     g-9 

'^  +  5     2;  + 10     2  +  4     ;2  +  9' 

Hint.     Simplify  each  member  before  clearing  of  fractions. 
lo>     — — —  ——^—  • 

x  +  2      x  +  'd      x-\-Q>      x-^7 

19    ^~^  I  -'^  —  '^  _  ^  —  5   ,  g;  — 3 
03  —  2     0?- 8     x  —  6     X—  4: 

0^  +  2     a^-2^10-2a^  2a;  +  l  8       ^2a;-l 

^°*    oj  +  l      ic-1       x'-l   '  '    2x-l      4.0^-1      2x4-1' 

21  1  + ^_  =  ^±^.  25     ^-^II^-?:  =  0. 

•    2     2(07  +  1)      cc  +  6  X        bx        a 

2  s       5  s- 3       1  ^^    2c  ,  6      c(a-2a;) 

22  —  •  =-•  26. \--=— -• 

3      10s2_l     3  a      x      a{2-x) 

23.       1      ^  6  (1  -  a?)  ^  -  g;  ^7.  ^  +  ^a^         =^-^. 

1  —  x  X  x  —  1  '    x^  —  cx-^ax  —  ac      c  —  x 

3v  15  ^10  g 

v  +  l     3v2  +  v-2     3v-2 

2 5  m     ^  m  +  29  _  ^ 

•    ^_5     3m  +  2      (m-5)(3m+2) 

^^    07  +  7  a,    x  —  a       x-^7  a       a  —  x 
30.    — --: h  — 


aj  +  6a     x  —  3a       x-\-a       2a-\-x 

a(b  —  x)      o{c  —  x)      a  {c  —  x) 

17+3     ,^18     21  _^     100^5 

„     «  .  X       X X 3 

^^-    -§-  +  — 5-      -9-  +  -T5— 


33.   If    C  represents  the  circumference  of  a  circle  whose  radius 
n 
is  Rf  then  - —  =  7r(cf.  Ex.  37,  p.  67)  ;  solve  this  equation  for  O; 
2  7? 

for  i?  .     Taking  tt  =  3|,  find  the  value  of  it  when  0—56. 


34.    v  =  -- 

t 

37.    II'=-ii'.s'. 

35.    at  =  v. 

38     F^^". 

36.    Z>  =  f 

39.    V  =u  —  gt. 

PROBLEMS 

07]  SIMPLE  EQrATlONS  183 

Solve  each  of  the  following  equations  for  each  letter  it  contains: 

40.  i(F-32)  =  a 

41.  s  =  ig(2t-l). 

42.  -  +  -=-• 
^      J?'      / 


1.  Three  fourths  of  a  certain  number  exceeds  |  of  it  by  25. 
What  is  the  number  ? 

2.  The  sum  of  a  certain  number,  its  half,  and  its  third  is 
.36.     Find  the  number. 

3.  If  f  of  a  certain  number  diminished  by  J  of  that  number 
equals  3  more  than  i  of  the  number,  what  is  the  number  ? 

4.  The  sum  of  two  numbers  is  18,  and  the  quotient  of  the  less 
divided  by  the  greater  is  \.     What  are  the  numbers  ? 

5.  Divide  the  number  32  into  two  parts  such  that  -^  of  the 
larger  shall  equal  ^  of  the  smaller. 

6.  Divide  the  number  80  into  two  parts  such  that  -|  of  the 
smaller  shall  exceed  ^  of  the  greater  by  2. 

7.  Divide  the  number  25  into  two  parts  such  that  the  square 
of  the  greater  shall  exceed  the  square  of  the  smaller  by  75. 

8.  Wliat  number  must  be  added  to  each  term  of  the  fraction 
YY  so  that  the  resulting  fraction  shall  be  equal  to  J? 

9.  If  a  certain  number  is  added  to,  and  also  subtracted  from, 
each  term  of  the  fraction  |,  the  first  result  exceeds  the  second 
by  i;  find  the  number.     How  many  solutions  has  this  problem  ? 

10.  B's  present  age  is  18  years,  which  is  |  of  A's  age ;  after 
how  many  years  will  B's  age  be  |  of  A's  age  ? 

11.  The  combined  cost  of  a  table  and  a  chair  is  $  11,  of  the 
table  and  a  picture,  $  14,  and  the  chair  and  the  picture  together 
cost  3  times  as  much  as  the  table.     What  is  the  cost  of  each  ? 

12.  Divide  a  line  28  inches  long  into  two  parts  such  that  the 
length  of  one  part  shall  be  |  that  of  the  other. 


184  HIGH  SCHOOL  ALGEBRA  [Ch.  IX 

13.  A  field  is  twice  as  long  as  it  is  wide,  and  increasing  its 
length  by  20  rods  and  its  width  by  30  rods  would  increase  its 
area  by  2200  square  rods.  What  are  the  dimensions  of  this  field 
(cf.  Exs.  23-24,  p.  65)  ? 

14.  An  orchard  has  twice  as  many  trees  in  a  row  as  it  has 
rows.  By  increasing  the  number  of  trees  in  a  row  by  2,  and  the 
number  of  rows  by  3,  the  whole  number  of  trees  will  be  increased 
by  126.     How  many  trees  are  there  in  the  orchard  ? 

15.  An  officer  in  forming  his  soldiers  into  a  solid  square,  with 
a  certain  number  on  a  side,  finds  that  he  has  49  men  left  over ; 
and  if  he  puts  one  more  man  on  a  side,  he  lacks  50  men  of  com- 
pleting the  square.     How  many  men  has  he? 

16.  A  boy  was  engaged  at  15  cents  a  day  to  deliver  a  daily 
paper,  with  the  added  condition,  however,  that  he  was  to  forfeit 
5  cents  for  every  day  he  failed  to  perform  this  service ;  at  the 
end  of  60  days  he  received  $  7.     How  many  days  did  he  serve  ? 

17.  A  man  was  hired  for  30  days  on  the  following  terms  :  for 
every  day  he  worked  he  was  to  receive  $  2.50  and  board ;  for 
every  day  he  was  idle  he  was  to  receive  nothing,  and  was  to  pay 
75  cents  for  board.  If  his  total  earnings  were  $49,  how  many 
days  did  he  work  ? 

18.  The  square  of  a  certain  number  is  diminished  by  9,  and  the 
remainder  is  divided  by  10,  giving  a  quotient  w^hich  is  3  greater 
than  the  number  itself.     Find  the  number  (two  solutions). 

19.  If  a  certain  number  is  subtracted  from  each  of  the  four 
numbers  20,  24,  16,  and  27,  the  product  of  the  first  two  remain- 
ders equals  the  product  of  the  second  two.     What  is  the  number  ? 

20.  Find  a  fraction  whose  numerator  is  greater  by  3  than  one 
half  of  its  denominator,  and  whose  value  is  f . 

21.  The  numerator  of  a  certain  fraction  is  less  by  8  than  its 
denominator,  and  if  each  of  its  terms  is  decreased  by  5,  its  value 
will  be  i ;  what  is  the  fraction  ? 

22.  What  principal  at  4%  interest  for  3  years  amounts  to 
f  784  (cf.  Ex.  12,  p.  61)  ?  Solve  the  same  problem  if  the  amount 
is  $  10,140. 


97]  SIMPLE  EQUATIONS  135 

23.  I  invest  $6000,  part  at  6  %,  part  at  5%,  thus  securing 
a  total  yearly  income  of  $  325 ;  how  large  is  each  investment? 

24.  A  gentleman  made  two  investments  amounting  together 
to  $  4330 ;  on  one  he  lost  5  % ,  on  the  other  he  gained  12  % .  If 
his  net  gain  was  $  251,  how  large  was  each  investment  ? 

25.  In  a  certain  quantity  of  gunpowder,  made  up  of  saltpeter, 
sulphur,  and  charcoal,  the  saltpeter  weighs  6  lb.  more  than  i  of 
the  whole,  the  sulphur  5  lb.  less  than  ^  of  the  whole,  and  the 
charcoal  3  lb.  less  than  ^  of  the  whole.  How  many  pounds  of 
each  constituent  does  this  gunpowder  contain  ? 

26.  A  boy  bought  some  apples  for  24  cents;  had  he  received 
4  more  for  the  same  sum,  the  cost  of  each  would  have  been  1 
cent  less.     How  many  did  he  buy  ? 

27.  Knowing  the  time  consumed  by  an  automobile  in  making 
a  run  of  a  given  number  of  miles,  how  can  you  find  the  average 
speed  ?  How,  from  the  distance  and  the  rate,  can  you  find  the 
time  ?  How,  from  the  rate  and  the  time,  can  you  find  the  dis- 
tance ?     Illustrate  your  answers  (cf .  Exs.  15-16,  p.  61). 

28.  A  tourist  ascends  a  certain  mountain  at  an  average  rate 
of  1^  miles  an  hour,  and  descends  by  the  same  path  at  an  aver- 
age rate  of  4^  miles  an  hour.  If  it  takes  him  6J  hours  to  make 
the  round  trip,  how  long  is  the  path  (cf .  Exs.  35-36,  p.  67)  ? 

29.  A  north-bound  and  a  south-bound  train  leave  Chicago  at 
the  same  time,  the  former  running  2  miles  an  hour  faster  than 
the  latter.  If  at  the  end  of  1|-  hours  the  trains  are  141  miles 
apart,  find  the  rate  of  each. 

30.  In  running  180  miles,  a  freight  train  whose  rate  is  f  that 
of  an  express  train  takes  2  hours  and  24  minutes  longer  than  the 
express  train.     Find  the  rate  of  each. 

31.  If  the  freight  train  of  Ex.  30  requires  6  hours  longer  than 
the  express  train  to  make  the  run  between  Buffalo  and  New  York, 
how  far  apart  are  these  two  cities  ? 

32.  An  express  train  whose  rate  is  40  miles  an  hour  starts 
1  hour  and  4  minutes  after  a  freight  train  and  overtakes  it  in 
1  hour  and  36  minutes.     Find  the  rate  of  the  freight  train. 


136  HIGH  SCHOOL  ALGEBRA  [Ch.  IX 

33.  An  automobile  runs  10  miles  an  hour  faster  than  a  bicycle, 
and  it  takes  the  automobile  6  hours  longer  to  run  255  miles  than 
it  does  the  bicycle  to  run  63  miles.     Find  the  rate  of  each. 

How  many  solutions  has  the  equation  of  this  problem  ?  Is 
each  of  these  also  a  solution  of  the  problem  itself  ? 

34.  A  steamer  now  goes  5  miles  downstream  in  the  same  time 
that  it  takes  to  go  3  miles  upstream,  but  if  its  rate  each  way  is 
diminished  by  4  miles  an  hour,  its  downstream  rate  will  be  twice 
its  upstream  rate.     What  is  its  present  rate  in  each  direction  ? 

35.  A  steamer  can  go  20  miles  an  hour  in  still  water.  If  it 
can  go  72  miles  with  the  current  in  the  same  time  that  it  can 
go  48  miles  against  the  current,  how  swift  is  the  current? 

Hint.  Let  x  =  the  rate  of  the  current  (in  miles  per  hour)  ;  then  20  —  jc  = 
the  steamer's  rate  upstream,  and  20  +  x  its  rate  downstream.     (Why  ?) 

36.  A  man  rows  downstream  at  the  rate  of  6  miles  an  hour, 
and  returns  at  the  rate  of  3  miles  an  hour.  How  far  downstream 
can  he  go  and  return  if  he  has  2^  hours  at  his  disposal  ?  At 
what  rate  does  the  stream  flow  ? 

37.  At  what  time  between  2  and  3  o'clock  are  the  hands  of  a 
clock  together  ? 

Hint.  Make  drawing,  or  use  model  of  clock  face.  Let  x  =  the  number 
of  minute  spaces  over  which  the  minute  hand  passes  after  2  o'clock  before 
the  two  hands  come  together ;  then  —  =  the  number  of  minute  spaces  over 

which  the  hour  hand  passes  in  the  same  time  (why?);   and  ic  =  -^  +  10. 
(Why?)  ^^ 

38.  At  what  time  are  the  hands  of  a  clock  together  between 
8  and  9  ?     between  5  and  6  ?     6  and  7  ?     11  and  12  ? 

39.  At  what  time  between  3  and  4  o'clock  is  the  minute  hand 
15  minute  spaces  ahead  of  the  hour  hand  ? 

40.  At  what  time  do  the  hands  of  a  clock  extend  in  opposite 
directions  between  4  and  5  ?    between  2  and  3  ?    7  and  8  ? 

41.  The  tens'  digit  of  a  certain  two-digit  number  is  ^  the  units' 
digit,  and  if  this  number,  increased  by  27,  is  divided  by  the  sum 
of  its  digits,  the  quotient  will  be  6^1^.  What  is  the  number 
(cf .  Prob.  4,  p.  64)  ? 


07]  SIMPLE  EQUATIONS  137 

42.  Divide  72  into  four  parts,  such  that  if  the  first  is  divided 
by  2,  the  second  multiplied  by  2,  the  third  increased  by  2,  and  the 
fourth  diminished  by  2,  the  results  will  all  be  equal. 

43.  M  can  do  a  certain  piece  of  work  in  8  days,  and  N  can  do 
it  in  12  days ;  in  how  many  days  can  the  two  do  it  when  working 
together  (cf.  Ex.  41,  p.  67)? 

44.  Two  plasterers,  A  and  B,  working  together,  can  plaster  a 
house  of  a  certain  size  in  12  days,  while  A,  working  alone,  can 
plaster  such  a  house  in  18  days.  In  how  many  days  can  B  alone 
do  the  work  ? 

45.  A  reservoir  is  fitted  with  three  pipes,  one  of  which  can 
empty  it  in  4  hours,  another  in  3  hours,  and  the  third  in  1 J  hours. 
If  the  reservoir  is  half  full,  and  the  three  pipes  are  opened,  in 
what  time  will  it  be  emptied  ? 

46.  The  first  of  three  outlet  pipes  can  empty  a  certain  cistern 
in  2  hr.  and  40  min.,  the  second  in  1  hr.  and  15  min.,  and  the 
third  in  2  hr.  and  30  min.  If  the  cistern  is  f  full,  and  all  three 
pipes  are  opened,  in  what  time  will  it  be  emptied  ? 

47.  A  can  do  a  piece  of  work  in  6  days,  and  B  can  do  it  in  14 
days.  A,  having  begun  this  work,  had  later  to  abandon  it ;  B  took 
his  place  and  finished  the  work  in  10  days  from  the  time  it  was 
begun  by  A.     How  many  days  did  B  work? 

48.  A  certain  number  is  increased  by  1,  and  also  diminished 
by  1 ;  it  is  then  found  that  twice  the  reciprocal  of  the  second 
result  minus  3  times  the  reciprocal  of  the  first  result  equals  \. 
What  is  this  number  ?     How  many  solutions  has  this  problem  ? 

49.  A  picture  whose  length  lacks  2  inches  of  being  twice  its 
width  is  inclosed  in  a  frame  4  inches  wide.  If  the  length  of  the 
frame  divided  by  its  width,  plus  the  length  of  the  picture  divided 
by  its  width,  is  3^,  what  are  the  dimensions  of  the  picture  ?  How 
many  solutions  has  the  equation  of  this  problem?  Is  each  of 
these  a  solution  of  the  problem  also  ? 

50.  A  gentleman  invested  ^  of  his  capital  in  4%  bonds 
(i.e.,  bonds  yielding  4  %  interest  per  annum),  f  of  it  in  S^  % 
bonds,  and  the  remainder  in  6  %  bonds,  purchasing  all  these  bonds 
at  par.     If  his  total  annual  income  is  $  3412.50,  find  his  capital. 


138  HIGH  SCHOOL  ALGEBRA  [Ch.  IX 

51.  At  what  time  between  9  and  10  o'clock  is  the  hour  hand 
20  minute  spaces  in  advance  of  the  minute  hand? 

52.  A  pedestrian  finds  that  his  uphill  rate  of  walking  is  3  miles 
an  hour,  and  his  downhill  rate  4  miles  an  hour.  If  he  walked  60 
miles  in  17  hours,  how  much  of  this  distance  was  uphill  ? 

53.  A  wheelman  and  a  pedestrian  start  at  the  same  time  for  a 
place  54  miles  distant,  the  former  going  3  times  as  fast  as  the 
latter ;  the  wheelman,  after  reaching  the  given  place,  returns  and 
meets  the  pedestrian  6|  hours  from  the  time  they  started.  At 
what  rate  does  each  travel  ? 

54.  In  a  mixture  of  water  and  listerine  containing  21  ounces 
there  are  7  ounces  of  listerine.  How  much  listerine  must  be 
added  to  make  the  new  mixture  J  pure  listerine  ? 

Hint.     Let  x  =  the  number  of  ounces  of  listerine  to  be  added.     Then 

7  +x 


(Wliy?) 


21  +rc     4 

55.  In  an  alloy  of  silver  and  copper  weighing  90  oz.  there  are 
6  oz.  of  copper ;  find  how  much  silver  must  be  added  in  order 
that  10  oz.  of  the  new  alloy  shall  contain  but  f  oz.  of  copper. 

56.  If  80  lb.  of  sea  water  contains  4  lb.  of  salt,  how  much 
fresh  water  must  be  added  in  order  that  45  lb.  of  the  new  solu- 
tion may  contain  1 J  lb.  of  salt  ? 

57.  If  a  mixture  of  water  and  alcohol  is  y%  pure  alcohol,  how 
much  water  must  be  added  to  one  gallon  of  the  mixture  to  make 
a  new  mixture  ^  pure  alcohol  ? 

58.  Solve  Prob.  57  if  the  given  mixture  is  80  %  pure  alcohol 
and  the  required  mixture  50  %  pure  alcohol. 

59.  How  much  alcohol  must  be  added  to  one  gallon  of  a  mixture 
40  %  pure  to  make  a  new  mixture  75  %  pure  ? 

60.  What  fractional  part  of  a  6  %  solution  of  salt  and  water 
(salt  water  of  which  6  %  by  weight  is  salt)  must  be  allowed  to 
evaporate  in  order  that  the  remaining  portion  of  the  solution  may 
contain  12  %  of  salt  ?     that  it  may  contain  8  %  of  salt  ?     10  %  ? 


07-98]  SIMPLE  EQUATIONS  139 

61.  A  physician  having  a  6%  solution  of  a  certain  kind  of 
medicine  wishes  to  dilute  it  to  a  3|  %  solution.  What  percent- 
age of  water  must  he  add  to  the  present  mixture  ? 

62.  If  the  specific  gravity  of  brass  is  8^,*  while  that  of  iron  is  7|-, 
and  if,  when  immersed  in  water,  57  lb.  of  an  alloy  of  brass  and  iron 
displaces  7  lb.  of  water,  find  the  weight  of  each  metal  in  the  alloy. 

63.  If,  on  being  irtimersed  in  water,  97  oz.  of  gold  displaces 
5  oz.  of  water,  and  21  oz.  of  silver  displaces  2  oz.  of  water,  how 
many  ounces  of  gold  and  of  silver  are  there  in  an  alloy  of  these 
metals  which  weighs  320  oz.  and  which  displaces  22  oz.  of 
water  ?     Find  the  specific  gravity  of  the  alloy ;  also  of  gold. 

98.  General  problems.    Formulas.    Interpretation  of  results. 

A  problem  in  which  the  known  numbers  are  represented  by 
letters,  instead  of  by  arithmetical  numerals,  is  often  called 
a  general  problem ;  it  includes  all  those  particular  problems 
which  may  be  obtained  by  giving  particular  values  to  these 
letters.     Some  problems  of  this  kind  are  given  below. 

Prob.  1.  A  yacht  was  chartered  for  a  pleasure  party  of  12,  the 
expense  to  be  shared  equally ;  3  members  of  the  proposed  party 
being  unable  to  go,  the  share  of  each  of  the  others  had  to  be 
increased  by  $  2.  How  much  was  paid  for  the  yacht  ?  How 
much  was  each  to  pay  under  the  original  arrangement  ? 

SOLUTION 

Let  X  =  the  number  of  dollars  each  member  was  to  have  paid, 
then  x-\-2  =  the  number  of  dollars  each  participant  did  pay ;  hence 
12  X  and  9  (a?  -f  2)  each  represent  the  number  of  dollars  charged 
for  the  yacht ; 

therefore  12  a;  ==  9  (a;  +  2), 

i.e.y  12  a;  =  9  a; +  18, 

and  therefore  •    x  =  6j  and  1 2  a;  =  72 ; 

hence  the  amount  each  was  to  have  paid  is  $  6,  and  the  rental  price 
of  the  yacht  is  $  72. 

*  This  means  that  a  given  volume  of  brass  weighs  8f  times  as  much  as  an 
equal  volume  of  water. 

HIGH  SCH.  ALG.  — 10 


140  BIGH  SCHOOL   ALGEBRA  [Cii.  IX 

Prob.  2.  Substitute  p,  q,  and  d,  for  12,  3,  and  2,  respectively, 
in  Prob.  1,  and  solve  the  problem  thus  formed. 

SOLUTION 

Let  X  =  the  number  of  dollars  each  member  was  to  have  paid, 
then  x-\-d  =  the  number  of  dollars  each  participant  did  pay;  hence 
2}x  and  (/)  —  q)  -  {x-\-d)  each   represent   the  number   of   dollars 
charged  for  the  yacht ; 
therefore  px  =z  {p  —  q)  {x  -\-  d)= px  -\- pd  —  qx  —  qd  ; 

whence  x  =     ^^^  ~  -^^ ,  the  amount  each  was  to  pay, 

and  px=:p  •     ^^~^\  the  rental  price  of  the  yacht. 

Remark.  The  solutions  of  Probs.  1  and  2  are  alike  except 
in  this:  In  the  solution  of  Prob.  1  the  numbers  given  in  that 
problem  (12,  3,  and  2)  have,  by  combining,  completely  lost  their 
identity  before  the  result  is  reached  ;  but  in  the  solution  of  Prob. 
2  the  given  numbers  {p,  q,  and  d)  preserve  their  identity  to  the  end. 

For  this  reason  the  result  in  Prob.  2  may  be  used  as  a  formula, 
by  means  of  which  the  answer  to  Prob.  1,  or  to  any  like  problem, 
may  be  immediatel}^  written  down. 

E.g.,  substituting  12,  3,  and  2  for  p,  q,  and  d  respectively,  in 
the  solution  of  Prob.  2,  gives  the  answer  to  Prob.  1. 

The  solution  of  Prob.  2,  therefore,  includes  that  of  Prob.  1. 
The  first  problem,  and  all  like  numerical  problems,  are  merely 
particular  cases  of  the  second,  which  is  called  a  general  problem. 

Prob.  3.  Divide  m  golf  balls  into  two  groups,  in  such  a  way 
that  the  first  group  shall  contain  n  balls  more  than  the  second. 

Solution.     Let     a?  =  the  number  of  balls  in  the  first  group. 
Then  m  —  x  =  the  number  of  balls  in  the  second  group, 

and,  therefore,  by  the  condition  of  the.  problem, 
x  =  m  —  x-\-n] 

whence  x  = -,  the  number  in  the  first  group, 


2 

m  —  x  =  m 
second  group, 


and  m  —  x  =  m ^~-  =  — — ^,  the  number  in  the 


98]  SIMPLE  EQUATIONS  141 

As  in  Prob.  2,  so  here,  the  general  solution  may  be  employed  to  solve  any 
particular  problem  of  the  same  kind.      For  example,  if  m  =  30  and  n  =  4, 

then  the  two  groups  contain,  respectively,  — ^^  and  — ^^^—  balls,  i.e.,  17  and 

18  ;  while,  if  w  =  10  and  w  =  2,  then  the  two  groups  contain  6  and  4  balls, 
respectively. 

If,  however,  m  =  10  and  n  =  14,  then  the  number  of  balls  in  the  two  groups, 

as  given  by  the  above  solution,  is  — ^ —  and  - — ^^- — ,  respectively,  i.e.,  12 

and  —  2  ;  but  since  there  cannot  be  an  actual  group  containing  —  2  golf 
balls,  therefore  this  last  problem  is  impossible,  and  the  impossibility  is  indi- 
cated by  the  negative  result. 

Eemark.  Some  problems  admit  of  negative  results,  and  some 
do  not,  just  as  some  problems  admit  of  fractional  results,  while 
others  do  not.  The  nature  of  the  things  with  which  any  particu- 
lar problem  is  concerned  will  always  make  it  evident  whether  or 
not  fractional  or  negative  solutions  are  admissible. 

Prob.  4.  Two  boys,  A  and  B,  are  running  along  the  same 
road,  A  at  the  rate  of  a,  and  B  at  the  rate  of  h,  yd.  per  minute ; 
if  B  is  m  yd.  in  advance  of  A,  and  if  they  continue  running  at  the 
same  rates,  in  how  many  minutes  will  A  overtake  B  ? 

Solution.     Let  i)j  =  the  number  of  minutes  that  must  elapse 
before  A  overtakes  B.     Then  by  the  conditions  of  the  problem, 
ax  =  hx  4-  m, 

whence  x  = ,  the  number  of  minutes  before 

A  overtakes  B.  ^  ~ 

As  in  the  two  previous  problems,  so  here,  the  general  solution  may  be 
employed  to  solve  any  particular  problem  of  the  same  kind. 

QO 

E.g.,\ia  =  280,  h  =  270,  and  m  =  90,  then  x  = — =  9;  i.e.,  A  will 

''  '  '  '  '  280 270 

overtake  B  in  9  minutes. 

Again,  if  a  =  280,  b  =  280,  and  m  =  90,  then  x  = =  '— ;  i.e.,  an 

'280  -  280       0 

infinite  number  of  minutes  will  elapse  before  A  overtakes  B  ;  in  other  words, 

A  will  never  overtake  B.     Compare  §  41  (iii),  also  Ex.  7,  p.  53. 

QO 

But  if  a  =280,  b  =  290,  and  m  =  90,  then  x  = — =  -9;  i.e.,  the 

280  -  290 

two  boys  are  together  —  9  minutes  from  the  moment  they  were  observed, 

i.e.,  the  two  boys  loere  together  9  minutes  ago. 

Let  the  pupil  show  that  this  interpretation  of  the  negative  result  accords 

fully  with  the  physical  cuuditions  of  the  problem. 


142  HIGH  SCHOOL  ALGEBRA  [Ch.  IX 

Prob.  5.  The  present  ages  of  a  father  and  son  are  respectively 
50  and  20  years ;  after  how  many  years  will  the  father  bfe  4  times 
as  old  as  the  son  ? 

Solution.  Let  x  =  the  number  of  years  from  now  to  the  time 
when  the  father's  age  shall  be  4  times  that  of  the  son.  Then,  by 
the  conditions  of  the  problem, 

50  +  a;  =  4(20  +  a;), 
whence  a?  =  —  10. 

This  means  that  10  years  ago  the  father's  age  was  4  times  the 
son's. 

N.B.  The  general  problem  of  which  Prob.  5  is  a  particular  case,  may  be 
stated  thus :  The  present  ages  of  a  father  and  son  are,  respectively,  m  and  n 
years  ;  after  how  many  years  will  the  father  be  p  times  as  old  as  the  son  ? 

EXERCISE  LXVII 

6.  The  sum  of  two  numbers  is  a,  and  the  larger  exceeds  the 
smaller  by  b.     What  are  the  two  numbers  ? 

7.  By  substituting  in  the  formula  obtained  from  the  solution 
of  Prob.  6  above,  solve  Probs.  6'  and  7,  p.  64.  Could  Prob.  16, 
p.  65,  be  solved  by  means  of  the  same  formula  ? 

8.  Is  Prob.  9,  p.  64,  a  particular  or  a  general  problem  ?  Why  ? 
Make  a  general  problem  which  shall  include  this  one  as  a  par- 
ticular case.  Solve  the  new  problem  and  thus  find  a  formula  by 
which  Prob.  9,  p.  64,  may  be  solved. 

9.  Answer  the  questions  in  Ex.  8  above,  supposing  them  to 
have  been  asked  with  regard  to  Probs.  4  and  12,  p.  133. 

10.  Which  of  the  following  admit  of  fractional  results  :  Probs. 
14,  15,  18,  p.  134;  Probs.  24-26,  p.  135? 

11.  Do  any  of  the  problems  mentioned  in  Prob.  10  above  admit 
of  negative  results  ?     Explain. 

12.  By  a  slight  change  in  the  wording  of  Prob.  5  above^  make 
an  equivalent  problem  whose  answer  shall  be  positive.  This  an- 
swer should  agree  with  the  interpretation  of  the  negative  result 
oriven  in  Prob.  5. 


98]  SIMPLE  EQUATIONS  148 

13.  By  slightly  changing  the  wording  in  the  last  particular 
case  under  Prob.  4  above,  make  an  equivalent  problem  whose 
answer  shall  be  positive. 

14.  What  principal  at  c  %  for  t  years  will  earn  i  dollars  simple 
interest  ?  By  substituting  in  your  answer,  find  the  principal 
when  c  =  5,  I  =  270,  i  =  3  ;  also,  when  c  =  3i,  i  =  224,  t  =  8. 

15.  A  father  is  now  m  times  as  old  as  his  son;  in  p  years,  the 
father's  age  will  be  n  times  that  of  the  son.  Find  the  present 
age  of  each.  Also  interpret  your  result  when  m  is  less  than  n. 
Is  p  positive  or  negative  in  this  case  ? 

16.  Solve  the  equation  of  Prob.  2  above  for  d,  and  then  find 
the  value  of  d  corresponding  to  p  =  12,  g  =  2,  ic  =  4.  May  d  be 
fractional  in  value  ?     negative  ?     Explain. 

17.  M  can  do  in  a  days  a  piece  of  work  which  N  can  do  in  6 
days.  In  how  many  days  can  they  do  it  when  working  together  ? 
Use  this  answer  to  solve  Prob.  43,  p.  137. 

18.  A  merchant  has  two  kinds  of  sugar  worth,  respectively,  a 
and  h  cents  a  pound.  How  many  pounds  of  each  kind  must  he 
take  to  make  a  mixture  of  n  pounds  worth  c  cents  a  pound  ? 

19.  How  many  solutions  has  Prob.  18  if  a  =  5  =  c?  ifa  =  6 
while  c  differs  from  a  ?  Does  the  answer  to  Prob.  18  show  these 
facts  [cf.  §  41  (iii)  and  (iv)]  ? 

20.  An  alloy  of  two  metals  is  composed  of  m  parts  (by  weight) 
of  one  to  n  parts  of  the  other.  How  many  pounds  of  each  of  the 
metals  are  there  in  a  pounds  of  the  alloy  ? 

21.  A  bell  made  from  an  alloy  of  5  parts  (by  weight)  of  tin  to 
16  of  copper,  weighs  4200  lb. ;  how  many  pounds  of  tin  and  of 
copper  in  the  bell  ?     How  is  Ex.  22  related  to  Ex.  21  ? 

22.  At  what  time  between  n  and  n  +  1  o'clock  will  the  hands 
of  a  clock  be  together  ?  By  means  of  your  answer  write  down 
the  answers  to  Prob.  38,  p.  136. 

23.  At  what  time  between  n  and  w  +  l  o'clock  will  the  hands 
of  a  clock  be  pointing  in  opposite  directions  if  n  is  less  than  6  ? 
if  n  is  greater  than  6  ?  if  n  equals  6  ?  By  means  of  your 
answer  write  down  the  answers  to  Prob.  40,  p.  136. 


CHAPTER  X 

SIMULTANEOUS  SIMPLE  EQUATIONS 

I.     TWO   UNKNOWN   NUMBERS 

99.  Indeterminate  equations.  A  simple  equation  in  one 
unknown  number  has  but  one  solution  (i.e.,  one  root,  cf. 
Chapter  IX),  but  an  equation  that  contains  two  or  more  un- 
known numbers  has  many  solutions. 

E.g.,  in  the  equation         3x+2y  =  6,  which,  when  solved 

for  y,  becomes  ^      o 

^'  b  —  ox 

we  see  that  if  the  values  1,  2,  3,  —  1,  etc.,  are  assigned  to  x,  then 
//  will  take  the  corresponding  values  1,  —  ^,  —  2,  4,  etc.  That  is, 
this  equation  is  satisfied  by  the  pairs  of  numbers : 

a;  =  11.  x  =  2     1.  »  =  3     1.   ^  =  -11.  ^^^ 

An  equation,  such  as  the  one  just  now  considered,  which 
has  an  infinite  number  of  solutions,  is  for  that  reason  called 
an  indeterminate  equation. 

EXERCISE    LXVm 

By  the  method  of  §  99  find  five  solutions  of  each  of  the  fol- 
lowing equations : 

1.  a; +  3 2/ =  7.  3.  5 a; +  3?/ =  11.             5.  2^y  =  5  +  32;. 

2.  a;  4-  2/  =  5.  4.  5  m  +  2  ti  =  15.            6.  v  —  vj  —  1. 

7.  How  many  solutions  has  each  of  the  above  equations  ? 
Why  ?     What  are  such  equations  called  ? 

8.  How  many  positive  integral  solutions  (i.e.,  solutions  in 
which  both  x  and  y  are  positive  integers)  has  the  equation 
3a;  +  22/=ll? 

Hint.     Solve  the  equation  for  y,  and  thus  show  that  x  cannot  exceed  3. 

144 


99-100]  SIMULTANEOUS   SIMPLE  EQUATIONS  145 

9.  By  the  method  of  Ex.  8  find  four  positive  integral  sohi- 
tions  of  the  equation  2x-\-y  =  9.  How  many  such  solutions  has 
this  equation  ? 

10.  If  possible,  find  positive  integral  solutions  of  the  equations 
in  Exs.  1-6  above. 

Show  that  the  following  have  no  positive  integral  solutions : 

11.  2x-4:y  =  l.         12.  3x  +  6y  =  5.         13.  9x-{-3y  =  17. 

14.  Find  three  solutions  of  the  equation  2x  —  5y  +  Sz  =  6', 
also,  three  solutions  of  the  equation  2  x  +  3y -\-4:Z  =  20. 

15.  A  farmer  spent  $22  in  purchasing  two  kinds  of  lambs,  the 
first  kind  costing  him  $3  each,  and  the  second  kind  $5  each. 
How  many  of  each  kind  did  he  buy  ? 

Hint.  Let  X  =  the  number  of  the  first  kind,  and  y  =  the  number  of  the 
second  kind ;  then  Sx  -\-  5y  =  22,  where  x  and  y  are  positive  integers. 

16.  A  man  spends  $300  for  cows  and  sheep  costing,  respec- 
tively, $4:5  and  $6  a  head;  how  many  of  each  does  he  buy? 

17.  In  how  many  ways  may  a  19-pound  package  be  weighed 
with  5-pound  and  2-pound  weights  ? 

18.  How  many  pineapples  at  25  cents  each,  and  watermelons 
at  15  cents  each,  can  be  purchased  for  $2.15? 

19.  Divide  a  line  which  is  100  feet  long  into  two  parts,  one  of 
which  shall  be  a  multiple  of  11  feet,  the  other  of  6  feet. 

20.  Find  the  least  number  which  when  divided  by  4  gives  a 
remainder  of  3,  but  when  divided  by  5  gives  a  remainder  of  4. 

100.    Simultaneous    equations.      Independent    equations.* 

The  equations  Sx-\-2i/  =  5 

and  x—2y  =  l^ 

have,  individually,  an  infinite  number  of  solutions  (cf .  §  99) ; 

they  also  have  07ie  solution,  viz.,  x=3  and  «/=  —2,  in  common ; 

I.e.,  these  values  of  x  and  y  satisfy  each  of  the  given  equations. 

xA  set  of  equations,  like  those  above,  having  one  or  more 

*  If  time  permits,  read  §§  137-140,  also  §  142,  in  connection  with  §§  100-101. 
This  plan  will  make  the  definitions,  and  also  the  operations,  more  concrete. 


146  HIGH  SCHOOL  ALGEBRA  [Ch.  X 

solutions  in  common,  is  usually  called  a  system  of  simul- 
taneous equations. 

Simultaneous  equations  are  often  called  consistent  equa- 
tions, while  two  equations  which  have  no  solution  in  com- 
mon are  called  inconsistent  equations.     Thus,  x-\-^  =  4:  and 

2  a;  -f  2  ^  =  9  are  inconsistent  equations. 

Two  or  more  equations,  no  one  of  which  can  be  derived 
from   the   others,  are  called   independent   equations.     Thus, 

3  a?+^  =  11  and  7  x  —  i/=9  are  independent  ;  but  3  a;-|-^  =  ll 
and  6  a;  H-  2  ?/  =  22  are  not  independent,  the  second  being 
obtained  by  multiplying  each  member  of  the  first  by  2. 

101.  Solving  simultaneous  equations.  The  solving  of  a 
system  of  simultaneous  equations  is  the  process  of  finding 
the  solutions  which  these  equations  have  in  common. 

x-\-y  =  4.,  (1) 


Ex.  1.    Solve  the  equations   , 

^  'x-y  =  2.  (2) 

Solution.  Adding  these  two  equations,  member  to  member 
(Ax.  1),  gives  2i.  =  6, 

whence  x  =  3. 

Substituting  this  value  of  x  in  Eq,  (1)  gives 

whence  2/  =  1- 

Moreover,  these  numbers,  viz.,  x  =  S  and  y  =  l,  when  substi- 
tuted in  the  given  equations,  check;  therefore  they  constitute  a 
solution  of  these  equations. 

3x-\-2y  =  26,  (1) 


Ex.  2.   Solve  the  equations    , 

^  '5x  +  9y  =  SS.  (2) 

Solution.     On  multiplying  both  members  of  Eq.  (1)  by  5,  and 
of  Eq.  (2)  by  3,  these  equations  become,  respectively, 

15aj-f-10i/  =  130,  (3) 

15ic  +  27?/  =  249;  (4) 

and  (Ax.  2)  subtracting  Eq.  (3)  from  Ex.  (4)  gives 

17  2/ =  119, 
whence  y=T, 


100-102]  SIMULTANEOUS  SIMPLE  EQUATIONS  l-iT 

Substituting  this  value  of  y  in  any  one  of  the  equations  con- 
taining both  X  and  y  gives 

a;  =  4; 
and  since  these  numbers,  viz.,  x  =  4  and  y  =  7,  check,  therefore 
they  constitute  a  solution  of  the  given  system  of  equations. 

102.  Elimination.  Any  process  of  deducing  from  two  or 
more  simultaneous  equations  other  equations  which  contain 
fewer  unknown  numbers  is  called  elimination.  Such  a  process 
eliminates  (i.e.,  gets  rid  of)  one  or  more  of  the  unknown 
numbers,  and  thus  makes  the  finding  of  a  solution  easier. 

That  particular  plan  of  elimination  which  was  followed  in 
the  examples  given  in  §  101  is  known  as  elimination  by 
addition  and  subtraction.  It  is  evident,  moreover,  that  this 
method  is  applicable  to  any  pair  of  such  equations.  The 
procedure  may  be  formulated  thus  : 

(1)  Multiply  the  given  equations  hy  such  numbers  as  will 
make  the  coefficient  of  the  letter  to  he  eliminated  the  same  (in 
absolute  value)  in  both  equations. 

(2)  Subtract  or  add  these  last  two  equations  (according  as 
the  terms  to  be  eliminated  have  like  or  unlike  signs). 

(3)  Solve  the  resulting  equation  for  the  unknown  number 
which  it  contains. 

(4)  Substitute  that  value  in  any  one  of  the  earlier  equations 
and  thus  find  the  other  unknown  number o 

(5)  Check  the  results. 

Note.  Number  (2)  above  is  permissible  only  because  the  letters  have  the 
same  value  in  both  equations  (of.  §  101). 

EXERCISE  LXIX 
Solve  each  of  the  following  systems  of  equations  and  check 
the  results : 


{2x 


4. 


-2/  =  5,  g     ja;  +  3y=ll, 

H-32/  =  17.  '    |3a;-4.y  =  7. 

=  15, 
=  33, 


|a;4-2?/  =  9,  {2v-\-du 

1 2 a; +  2/ =  15,  '    [4^  +  9?^ 


148  HIGH  SCHOOL  ALGEBRA  [Ch.  X 


3a;  +  7?/  =  6.  [12x-9y  =  0. 

2x-\-5y=:S,  f5s  +  6f  =  17, 

8.    ^       _  _  .  _  11. 


7  a: +  10  2/ =-17.  [6.9  +  5^  =  16. 

15  oj  +  77  2/  =  92,  f  4  m  - 15  7i  =  32, 

9.    i  12.     ' 


5aj-32/=2.  [10m-9w  =  -34. 

13.  What  is  meant  by  saying  that  two  equations  are  simul- 
taneous ?  consistent?  inconsistent?  independent?  Show  the 
appropriateness  of  these  terms. 

14.  If  in  two  simultaneous  equations  the  coefficients  of  the 
letter  to  be  eliminated  are  prime  to  each  other  (cf.  Ex.  11),  what 
is  the  simplest  multiplier  for  the  first  equation  ?  for  the  second  ? 
Answer  the  same  questions  when  the  coefficients  under  considera- 
tion are  7iot  prime  to  each  other  (cf.  Ex.  12). 

Solve  the  following  systems  of  equations  and  check  the  results  : 

5p-{-3q=6S,  {3m-2n  =  7, 

15.  r.         .  '  19. 


2^  +  5^  =  69.  [4  m  — 7w=— 47. 

22  .T- 8  ^  =  50,  (4r-f  5s=-19, 


26x-\-6y  =  175.  '     [2  r-\-3  s= -10^-^. 

15  it-  -f  14  ?/  =  -  45,  f  35  a;  -  27  ?/  =  -  19, 


25  a^  -  21 2/  =  -  75.  [  21 2/  -f-  40  a;  =  82 

18  t^  +  10  V  =  59,  f  28  a;  -  23  y  =  33, 

18.    <!  _         _         _  22.    '  -^ 


12w-15v  =  28i.  [63  a; -25  2/ =  199. 

103.  Other  methods  of  elimination.  Besides  the  method  of 
elimination  described  in  §  102,  there  are  several  other  methods 
that  serve  the  same  purpose  ;  two  of  these,  which  are  often 
useful,  will  now  be  explained. 

(i)  Elimination  by  substitution. 

3x-Ay  =  7,  (1) 


Ex.  1.  Solve  the  system  of  equations    ,  ^        ^  ^ 

^  ^  '2x  +  3y  =  16.  (2) 


102-108]  SIMULTANEOUS   SIMPLE  EQUATIONS  149 

SOLUTION 

From  Eq.  (1)  x  =  ^^^;  [§  99 

o 

on  substituting  this  expression  for  cc,  Eq.  (2)  becomes 

2(^)  +  32,=  16,  (3) 

whence  (§  94)  2/  =  2; 

on  substituting  this  vahie  in  either  Eq.  (1)  or  Eq.  (2),  we  obtain 

X  =  r). 
Moreover,  these  values,  viz.,  x  =  5  and  y  =  2,  check  ;    therefore, 
they  constitute  a  solution  of  the  given  system  of  equations. 

This  method  of  elimination  is  known  as  elimination  by 
substitution ;  it  is  manifestly  applicable  to  any  such  system 
of  equations  as  the  above. 

The  student  may  solve,  by  this  method,  the  system 

I  8  M  -  4  u  =  19, 

1  5  w  +  2  V  =  10, 
being  careful  to  check  the  result,  and  then  write  out  a  "rule"  for  applying 
this  method  to  all  such  exercises. 

(ii)  Elimination  by  comparison. 

r.  .        .  n  .  f3a;-42/  =  7,  (1) 

Ex.  2.    Solve  the  system  of  equations   < 

-^  ^  \2x-{-3y  =  16.  (2) 

SOLUTION 

From  Eq.  (1),  x  =  I^til,  and  from  Eq.  (2),  x=  l^Llzll .     Now, 

since  a?  is  to  have  the  same  value  in  each  of  these  equations, 
therefore  Tj^^lfi-l^.  ^3^ 

Solving  Eq.  (3)  gives       y  =  2, 
whence,  substituting  this  value  in  either  of  the  given  equations, 

X  =  5. 

Moreover,  these  values,  viz.,  x  =  5  and  y  =  2,  check ;  therefore, 
they  constitute  a  solution  of  the  given  system  of  equations. 

This  method  of  elimination  is  called  elimination  by  com- 
parison ;  it  is  applicable  to  all  such  systems  of  equations. 


150  niGll   SCHOOL  ALGEBBA  [Ch.  X 

The  student  may  solve,  by  this  method,  the  system 
I    8r  +  5s  =  3, 
ll2r-7s=48, 
and  then  write  out  a  "  rule  "  for  applying  this  method  to  all  such  exercises. 

EXERCISE   LXX 

Solve  the  following  systems  of  equations,  using  first  elimina- 
tion by  substitution,  and  then  that  by  comparison ;  observe  which 
method  is  easier  in  the  different  exercises : 

•      5a;-2^  =  10. 


4.    ^  -  7. 


llx-lOy 

=  14, 

ox+1 y= 

41. 

21 2/ H- 20  a; 

=  165, 

772/-30«^ 

=  295. 

8  ^  - 10  y  = 

14, 

6^  +  35 'y  = 

:41. 

4x  +  2/  =  34, 
42/  +  a^  =  16. 

^    |2a^  +  72/  =  34, 
■    l5a;  +  92/  =  51. 

9.  Using  the  method  of  elimination  by  comparison,  solve  each 
of  the  systems  of  equations  in  Exs.  3-6,  p.  147. 

10.  Using  the  method  of  elimination  by  substitution,  solve  each 
of  the  systems  of  equations  in  Exs.  7-10,  p.  148. 

11.  Show  that  elimination  by  comparison  is  merely  a  special 
case  of  elimination  by  substitution. 

Solve  Exs.  12-21  below ;  use  the  simplest  method  of  elimination 
in  each  case,  giving  reasons  for  your  choice  of  method ;  and  check 
your  results  as  the  teacher  directs  : 

7ic  +  42/  =  l,  r8?/-21v  =  5, 

±2.   {  ^         ,  '  15.    i  ' 

9a;  +  42/  =  3.  [6ii  +  14'y  =  -26. 

^^       3a;  +  52/  =  19,  ^^    |34a^-152/  =  4, 

5a;-42/  =  7.  *    [  51  a; +  25  2/  =  101.  • 


14. 


f  aj  - 11  ?/  =  1,  I  39  .'c  -  1 5  2/  =  93, 

jlll7/-9a?  =  99.  ^^'    [6535  +  17 2/  =  113. 


103-104]  SIMULTANEOUS  SIMPLE  EQUATIONS 


151 


18. 


19. 


19s  +  85^  =  350, 
17  s +  119  ^  =  442. 

j8.9-llw  =  0, 
[258-17  w;  =  139. 


20. 


21 


I 


3a;-ll^  =  0, 
19a;-19  2/  =  8. 

aj  +  9  2/+42  =  0, 
25  2/  +  a;  4- 20  =  0. 


104.  Equations  containing  fractions.  The  solution  of  si- 
multaneous equations  containing  fractions  is  illustrated  by 
the  following  examples : 

-2 


Ex.  1.    Given 


3 


13__2/ 


4-¥  =  4i 


;  to  find  a?  and  y. 


Solution.     On  multiplying  these  equations  by  12  and  6,  re- 
spectively, and  collecting  terms,  we  obtain 

4  a;  +  3  2/  =  29, 
and  3  ic  +  4  2/  =  27 ; 

whence  (§  101)  x  =  5  and  y  =  S. 

Moreover,  when  substituted  in  the  given  equations,  these  values 
check ;  they  are,  therefore,  the  solution  of  those  equations. 


Ex.  2.   Given 


3  s      r     r  (? 

1-5  =  0. 


16 


3s) 


> ;  to  find  r  and  s. 


Solution.     On  multiplying  these  equations  by  r  (r  —  3  s)  and 
3,  respectively,  they  become     r  +  4  (r  —  3  s)  ==  16, 
and  r_3_3s  =  0; 

whence  (§  101)  r  =  4  and  s  —  \. 

When  substituted  in  the  given  equations,  these  values  check ; 
they  are,  therefore,  the  solution  sought. 


Ex.  3.    Given 


1  +  1  =  3 
?-?  =  ! 


to  find  u  and  v. 


162 


HIGH   SCHOOL   ALGEBRA 


[Cn.  X 


Solution.     Instead  of  clearing  of  fractions  here,  it  is  better 

to  treat  -  and  -  as  the  unknown  numbers ;  we  may  even  substi- 

tute  a  single  letter  for  each  of  these  unknown  fractions. 

Thus,  on  substituting  x  for  -  and  y  for  -,  the  given  equations 

u  V 


become 

x  +  y  =  3, 

and 

2x-3y=l, 

respectively. 

Whence 

x  =  2  and  y  =  lf 

[§103 

i.e., 
whence 

i  =  2and  ^  =  1; 

U                       V 

u  =  ^  and  v  =  l', 

[§97 

and  these  values  are  found  to  check. 


EXERCISE  LXXI 


Solve  the  following  systems  of  equations^  and  check  the  results  ; 
eliminate  before  clearing  of  fractions  when  practicable : 


4. 


7. 


M=i^' 


4      2 

3  +  3        ' 


2. 


X      y 
6      2 


6i. 


-4-^-7 
2  +  3"    ' 

3^4 


+  5;2  =  -4, 


10. 


11. 


^^  +  5  =  3, 
X      y 

?-?  =  l 

X      y 

s     r 

^-?  =  7. 

s      r 

2r  +  3t  ,   ^+6 


5  7 

2r  —  bt  ,  r  +  7 


3 

h-2 
3 

2/^-7 


4 


=  2, 
=  1. 


3 

13-J 
0 


=  0, 


10. 


104j 


BIMULTANEOUS  SIMPLE  EQUATIONS 


153 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


3x-{-2y-\-6_. 
4:x-2y           ' 

3-7.v_2 

2ic  +  l       • 

8                  15 

y 

20.  . 

21.  . 

llr     5^     ^o 
|l2        8=^^- 

r3        6  _17 
2.T"^5.y     40' 
7        4        11 
2x     5  ;y      120 

5x  +  16y     3x—4: 

Sy-2x  =  7. 

n 


0, 


2w-5     3m-7 


2n-3     3m  +  l 
r3a;-2y  +  |_16 


a; -2/ 
15  +  y  — 2a;_g 
4a7— 5?/— 2 

2a;H-i/-50  =  0, 

?  =  ?l4-3. 

4  3 

5  +  ?  =  20, 
a;     y 

^  +  ^  =  10. 

a;     i/ 


3' 


2v     IV 


=  -3, 


:|-  +  -  =  23. 

2^      w 


22. 


23 


24. 


25. 


26. 


-4s  +  i^-?  =  0. 

7«     7 


|-i(2/-2)=K^-3), 
^-i(2/-l)=i(^-2). 

0, 


^     +     ' 


x-2     3-y 
x-1  _2y +  11 


6  5.5 

.2y  +  .5^.49a;-.7 

1.5  4.2      ' 
.5  a;-. 2  ^41      1.5.^ 

1.6  16 


11 


27. 


'5«  +  6: 

/  +  13_ 

3 

42/-2 

.'K  +  e 

2' 

13 

1  x—y 

_       2i 

x-y- 

■3-  . 

r    1 

2 

1 

4  ?,i  +  V 

u      —  u  (4  u 

-f-v)' 

-8 

v       2 

1^       1 

5 

1^— V      I 

^(2^^- 

-V) 

154 


HIGH  SCHOOL  ALGEBRA 


[Cii.  X 


28. 


29. 


30. 


31. 


32. 


7;  +  i  (3  V  -  w  -  1)  =  J-  +  f  (to  - 1), 
^(4^  +  3^(;)  =  3-V(7^o+24). 

2^-x  2       ' 

^ - |f^  =  ^^  +  3^-^    ^(.^  -g^  ^^^  p  ^3^^ 

s      o_5i  +  2s,s  —  3 

2~^-~t::7'"^~2~^ 


2t-^s 


«+l  ' 

22,. 

A  or  A- 

a; 

2 

'  17- 

-3a; 

50         2'- 

-1 

+  ^  =  12--^ 


-2^ 


2 
16  a;  -f  19 


3(^-2) 


=  8?/  + 


147  -  24  y 


U2x  I  3^   I    ^^  +  ^y    -31   I   3a;  +  4 

8t/  +  7      6a;-3.i/^^     4y-9 
10      "^2(2/-4)        "^      5 


J^J.gf.,  given 


;  to  find  X  and  y. 


105.    Literal  equations.     Literal  equations  may  be  solved 
by    the   methods   already   employed    in    solving    numerical 

equations. 

ax-\-by  —  c 

hx  -^ky=  I 

Solution.     On  multiplying  the  first  of  these  equations  by  k 
and  the  second  by  b,  they  become 

aJcx  +  hky  =  cJc, 
and  bhx  4-  bky  =  bl. 

Subtracting  member  from  member,  we  obtain 
akx  —  bhx  =  ch  —  bl^ 
i.e. ,  (oik  —  bh)x  =  ck  —  bl\ 

ck  —  bl 


whence 


ak  —  bh 


104-105]  SIMULTANEOUS   SIMPLE  EQUATIONS 


ir,5 


If  we  multiply  the  first  of  the  given  equations  by  h,  and  the 
second  by  a,  and  subtract,  we  eliminate  x,  and  find 

ch  —  al 

y  = • 

hh  —  dk 

Moreover,  these  values  of  x  and  y  check,  and  are,  therefore,  a 
solution  of  the  given  equations. 


EXERCISE  LXXII 

Solve  the  following  systems   of  equations  and  check  the  re- 
sults ;  eliminate  without  clearing  of  fractions  where  practicable : 

I  ax  -f  hy  =  m, 


1. 


2. 


3. 


4. 


x  +  y  =  c, 
x  —  y  =  d. 
ax  =  by, 
x-{-y  =  ab. 
'y-\-az  =  0, 
by  +  z=^l. 

X      y 


+  1-2  =  0, 
a     b 

'  —  ay  =  0. 

a     b 

^  +  JL  =  a  +  b. 
ab     ab 


[Hint.    Let  s  =  - and«  =  ^; 
a  0 

cf.  Ex.  3,  p.  152]. 


7. 


X      y 

^4-^  =  1. 

X     y 

HIGH    8CH.    ALG. 


10. 


11. 


12. 


13. 


14. 


15. 


I  bx  +  ay  =  n. 
x  —  y=a  —  b, 
ax-\-by  =  a^  —  b^. 

0, 


x-{-y     x  —  y 

a  b 

X  -\-y  _  x  —  y 
a 
1 
c 
1 
b 


b 
a     b 
X     y 
c  _a 
X      y 


1. 


ax     by     & 

bx     cy     0? 

(a  +  &)aj4-(«+c)2/: 

=  a+6. 

(a+c)a;4-(a+%  = 

=  a-\-c. 

a;  +  l      a  +  &  +  l 

^z  +  l^a-ft  +  l' 

x-y=^2b. 

'hx  +  lcy  =  4.h\ 

1      .      1     _ 

h 

x—k     y  —  h     k(y  —  h) 


11 


150  HIGH  SCHOOL   ALGEBllA  [Cii.  X 

16.  Under  what  circumstances  has  Ex.  8  above  no  finite  solu- 
tion ?  Explain  [cf.  §  41  (iii)].  Answer  this  question  with  regard 
to  Ex.  9  also ;  and  with  regard  to  Ex.  3. 

II.     THREE   OR  MORE  UNKNOWN   NUMBERS 

106.  Equations  containing  more  than  two  unknown  num- 
bers. The  methods  already  emploj^ed  in  the  solution  of 
systems  of  equations  containing  two  unknown  numbers 
(§§  101-105)  are  easily  extended  to  systems  containing  three 
or  more  iinknoAvn  numbers. 
Thus,  to  solve  the  system  of  equations 

r        a^  +  32/-«  =  5,  (1) 

3a^-j-62/  +  2^  =  3,  (2) 

[2x-3y-2z=^e>,  (3) 

we   first  eliminate  some  one  of   the  unknown   numbers,  say  2;, 

between  (1)  and  (2),  then  eliminate  the  same  unknown  number 

between  (1.)  and  (3);  in  this  way  we  obtain  two  new  equations, 

each  containing  the  two  unknown  numbers  x  and  y.     On  solving 

these  two  equations  we  find  x  and  y,  and  substituting  their  values 

in  (1)  we  find  z,  which  completes  the  solution  of  the  given  system. 

r    x  +  ?.y-    z  =  r,,  (1) 

Ex.  1.    Given  \Zx  +  (Sy -{-2z  =  ^,  (2) 

[2x-^y-3z=:Q',  (3) 

to  find  X,  2/,  and  z. 

SOLUTION 

Adding  2  times  Eq.  (1)  to  Eq.  (2),  member  to  member,  gives 

5x  +  12y  =  l^,  (4) 

and  subtracting  Eq.  (3)  from  3  times  Eq.  (1)  gives 

a^-fl2y  =  9.  (5) 

Now  subtracting  Eq.  (5)  from  Eq.  (4)  gives 

4  a!  =  4, 
whence  x  —  1. 

On  substituting  this  value  of  x  in  Eq.  (5),  we  obtain  2/  =  |;  and, 
with  these  values  of  x  and  y  in  Eq.  (1),  we  obtain  z  =  —  2. 

Moreover,  these  values  of  x,  y,  and  z  check ;  therefore  they 
constitute  a  solution  of  the  given  system  of  equations. 


lOo-luej  SIMULTANEOUS   SIMPLE  EQUATIONS  157 

Note.  Had  the  given  system  coiisisttd  of  four  equations,  containing  four 
unknown  numbers,  the  same  method  of  solution  would  still  have  sufficed. 
For,  by  eliminating-  some  one  of  the  unknown  numbers,  say  x,  between  (1) 
and  (2),  (1)  and  (3),  and  (1)  and  (4)  in  turn,  we  should  have  obtained  a 
system  of  three  equations  containing  the  remaining  three  unknown  numbers, 
which  could  then  have  been  solved  as  in  Ex.  1.  And  the  vakies  of  these  three 
unknown  numbers,  being  substituted  in  any  one  of  the  given  equations,  would 
have  determined  the  value  of  the  remaining  unknown  number. 

Similarly,  a  system  consisting  of  five  equations  containing  five  unknown 
numbers  can,  by  eliminating  some  one  of  these,  be  made  to  depend  upon  a 
system  of  four  equations  in  four  unknown  numbers ;  and  so  in  general  (see 
also  §  107). 

(2x-3y-2z==-l,  (1) 

Ex.  2.   Given  1 3 .« +  2;  =  6,  (2) 

l-c-\-y  +  z  =  S;  (3) 

to  find  the  values  of  x,  y,  and  z. 

SoLUTiox.  Since  the  second  of  these  equations  is  already  free 
from  the  unknown  number  y,  therefore  it  is  best  to  combine  Eqs. 
(1)  and  (3)  so  as  to  eliminate  y,  and  thus  obtain  another  equation 
involving  only  x  and  z.  On  adding  Eq.  (1)  to  three  times  Eq.  (3) 
we  obtain 

6x  +  z='6,  (4) 

and  on  subtracting  Eq.  (2)  from  Eq.  (4),  we  obtain 

2a?  =  2, 
whence  a;  =  l.  (5) 

On  substituting  this  value  of  x  in  Eq.  (2),  we  obtain 

2^=3; 
and  on  substituting  these  two  values  in  Eq.  (3),  we  obtain 

Moreover,  these  values  of  x,  y,  and  z,  viz.,  1,-1,  and  3,  check, 
and  therefore  constitute  a  solution  of  the  given  equations. 

EXERCISE   LXXIH 

Solve  each  of  the  following  systems  of  equations: 

(2x-\-  3  //  +  4  :^  =  20,  Ux-y-z  =  5, 

3.    J3.r  +  4//-f-r);^  =  26,  4.    hx-Ay-\-16  =  6z, 

[3x  +  oy-}-(Jz  =  'Sl.  [3y  +  2(z  —  l)  =  x. 


158 


HIGH   SCHOOL  ALGEBRA 


[Cii.  X 


5. 


6. 


7. 


8. 


9. 


10. 


11. 


12. 


13. 


7x  +  3y-2z  =  W, 
2x-^5y-\-Sz  =  3d, 
5x  —  y-\-5z  =  31. 

5x  —  6y-\-4:Z  =  lo, 
7x-\-Ay-3z  =  19, 
2x-\-y-^6z  =  4.6. 
2x-{-Ay-\-5z  =  19, 
-3x-\-5y-{-7z  =  S, 
Sx-3y-\-5z  =  23. 

5x-{-6y-12z  =  5, 

2x-2y-6z  =  -l, 
4:X  —  5y  +  3z  =  7^. 

y^z-S6  =  72-5x, 
93-ix-ly  =  ^^y-2z, 
ix-^ly-\-lz  =  5S. 

ix-{-ly  =  12-iz, 
iy  +  lz==8-\-lx, 

2x-5y-\-19  =  0, 
3y-4:Z  +  7  =  0, 
2z-5x-2  =  0. 

5     4"^5~   ' 
4     3^2 

X      y 


1  +  1  =  8. 

2  a; 


10, 


14. 


15. 


16. 


17. 


18. 


.S       2      1 

X     y     z 

1+^=1. 


x  +  z  =  3a-\-h, 

x-\-3y  =  ^c, 

y-{-2z  =  x, 
y-{-z=^x-2d. 

s  +  e/i+l'j^o, 

X     yj 

0, 

u-3(!.l)=.. 

a^y   ^1 

x-\-y     a 

yz        1 


5-2(1  +  1 

y    ^ 


19. 


2/  +  2;      5' 
xz    _\ 

x-{-z     c 

[Hint.   If  ^^  =  1,  then 

x  +  y     a 

xy  y     X 

(2v-{-Sx  +  y-z  =  0, 
3y-2x-\-z-4:V  =  21, 
2z  —  3v  —  y-\-x  =  6, 
v+4:X-^2y-3z  =  12. 


IOC] 


SIMULTANEOUS  SIMPLE  EQUATIONS 


159 


20. 


v-}-y  +  z  =  17, 

V  -\-x  +  z  =  16. 

Hint.     Adding  these   equations 
and  dividing  the  sum  by  3  gives 

'y-\-z-\-v  —  x  =  22, 
z-\-v-^x  —  y  =  lS, 
V -\- X -{- y  —  z  =14:, 
x  +  y-\-z  —  v  =  10. 


21. 


22. 


23. 


y-\-z  —  Zx  =  2a, 

z-v-3y  =  2b, 

y-\-x  —  3z  =  2cy 

l2v-{-2y  =a-b. 

(3u-\-5v-2x  +  Sz  =  2, 
2u-{-4:X-3y  —  z  =  3, 

u-{-v-\-z  =  2, 
(Sy-\-4iV  +  u  =  2y 
5  2;  H- 4  a;  —  7  v  =  0. 


PROBLEMS 

[Leading  to  simultaneous  equations  in  two  or  more  unknown  numbers.] 

1.  Find  two  numbers  whose  difference  is  3^5  of  their  sum,  and 
such  that  5  times  the  smaller  minus  4  times  the  larger  is  39. 

SOLUTION 


Let 
and 

Then,  by  the  conditions  of  the  problem 


X  =  the  larger  number, 
?/  =  the  smaller  number. 


x-y  = 


~35"' 


and  5  y  —  4  aj  =  39. 

Solving  these  equations,  we  obtain 

X  =  54:  and  y  =  51 ; 
and  these  numbers,  which  constitute  a  solution  of  the  equations  of  the  prob- 
lem, also  satisfy  the  problem  itself,  and  are,  therefore,  the  numbers  sought. 

2.  Find  two  numbers  snch  that  3  times  the  greater  exceeds 
twice  the  less  by  29,  and  twice  the  greater  exceeds  3  times  the 
less  by  1. 

3.  A  lady  purchased  20  yd.  of  gingham,  and  50  yd.  of  linen, 
for  $  29 ;  she  could  have  purchased  30  yd.  of  gingham,  and  20  of 
linen,  for  $  16.    What  was  the  price  of  each  material  ? 

4.  If  A's  money  were  increased  by  $  4000,  he  would  have  twice 
as  much  as  B.  If  B's  money  were  increased  by  $5500,  he  would 
have  3  times  as  much  as  A.     How  much  money  has  each  ? 


ir>o 


UIGIl  SCHOOL   ALGEBRA 


[Ch.  X 


5.  One  eleventh  of  A's  age  is  greater  by  2  years  than  1-  of  B's, 
and  twice  B's  age  equals  what  A's  age  was  13  years  ago.  Find 
the  present  age  of  each. 

6.  ABC  represents  a  triangle  whose  perimeter  is 
82  inches.  If  AB  =  BC  and  7  5(7=  17  AC,  find  the 
length  of  each  side  of  the  triangle. 

7.  A  man  having  $45  to  distribute  among  a  group 
of  children,  finds  that  he  lacks  $1  of  being  able  to 
give  $  3  to  each  girl  and  $  1  to  each  boy,  but  that  he 
has  just  enough  to  give  $2.50  to  each  girl  and  $1.50 
to  each  boy.    How  many  boys  and  how  many  girls  are 

there  in  this  group  ? 

8.  John  said  to  James,  "  Give  me  8  cents  and  I  shall  have  as 
much  as  you  have  left."  James  said  to  John,  "  Give  me  16  cents 
and  I  shall  have  4  times  as  much  as  you  have  left."  How  much 
money  had  each  ? 


C  9.   ABCD  represents  a  flower  bed  in  which 

BC  =  ^  AB.     If  the  perimeter  of  the  bed  is 
40  feet,  find  the  length  of  each  of  its  sides. 


10.  A  pound  of  tea  and  6  lb.  of  sugar 
together  cost  $  .96 ;  if  sugar  were  to  advance 
50%,  and  tea  10%,  then  2  lb.  of  tea  and 

12  lb.  of  sugar  would  cost  $2.28.     Find  the  present  price  of  tea, 

and  also  of  sugar. 


11.   A  grain  dealer  sold  to  one  customer  5  bushels  of  wheat, 
2  of  corn,  and  3  of  rye,  for  $6.60;  to  another,  2  of  wheat,  3  of 
corn,  and  5  of  rye,  for  $5.80;  and  to  another,  3  of 
wheat,  5  of  corn,  and  2  of  rye,  for  $5.60.     What  was 
the  price  per  bushel  of  each  kind  of  grain  ? 

12.  The  perimeter  of  the  triangle  CDE  is  68  in. ; 
four  times  CE  equals  CD  increased  by  four  times  DE, 
while  twice  CE  equals  DE  increased  by  twice  CD. 
How  long  is  each  side  of  the  triangle? 


106]  SIMULTANEOUS   SIMPLE  EQUATIONS  161 

13.  Divide  800  into  three  parts  such  that  the  first,  plus  -J-  of 
the  second,  plus  |-  of  the  third,  shall  equal  the  second,  plus  f  of 
the  first,  plus  J  of  the  third  :  each  of  these  sums  being  400. 

14.  Divide  90  into  three  parts  such  that  ^  of  the  first,  plus  -| 
of  the  second,  plus  i  of  the  third,  shall  be  30  ;  while  the  first  part 
increased  by  twice  the  second  shall  equal  twice  the  third. 

15.  A  boy  spent  $  4.10  for  oranges,  buying  some  at  the  rate  of 

2  for  5  cents,  some  at  3  for  10  cents.  Later  he  sold  all  at  4  cents 
apiece,  thereby  clearing  $  1.58.  How  many  of  each  kind  did  he 
buy  ? 

16.  If  a  certain  rectangular  floor  were  2  ft.  broader  and  3  ft. 
longer,  its  area  would  be  increased  by  64  sq.  ft. ;  but  if  it  were 

3  ft.  broader  and  2  ft.  longer,  its  area  would  be  increased  by  68 
sq.  ft.     Find  its  length  and  breadth. 

17.  Three  rectangles  are  equal  in  area ;  the  second  is  6  meters 
longer  and  4  meters  narrower  than  the  first,  and  the  third  is  2 
meters  longer  and  1  meter  narrower  than  the  second.  What  are 
the  dimensions  of  each  ? 

18.  The  sum  of  the  ages  of  a  father  and  son  will  be  doubled  in 
25  years ;  the  difference  of  their  ages  20  years  hence  will  just  equal 
^  of  their  sum  at  that  time.     Find  the  present  age  of  each. 

19.  A  merchant  sold  to  Mrs.  A.  2  yd.  of  cambric,  4  of  silk,  and 
3  of  flannel,  for  $5.05,  and  to  Mrs.  B.,  4  yd.  of  cambric,  5  of 
flannel,  and  2  of  silk,  for  $4.30.  If  2  yd.  of  flannel  cost  10  cents 
more  than  2  yd.  of  cambric  and  -J  yd.  of  silk  combined,  find  the 
price  of  each  per  yard. 

20.  The  tickets  to  a  concert  were  50  cents  for  adults  and  35 
cents  for  children.  If  the  proceeds  from  the  sale  of  100  tickets 
were  $39.50,  how  many  tickets  of  each  kind  were  sold  ? 

Solve  this  problem  also  by  using  but  one  letter  to  represent  an 
unknown  number. 

21.  Find  three  numbers  such  that  the  sum  of  the  reciprocals  of 
the  first  and  second  is  y\,  the  sum  of  the  reciprocals  of  the  first 
and  third  is  -f-g,  and  the  sum  of  the  reciprocals  of  the  second 
and  third  is  ||. 


162  HIGH  SCHOOL   ALGEBRA  [Cii.  X 

22.  The  sum  of  the  reciprocals  of  three  numbers  is  34 ;  the  re- 
ciprocal of  the  second  minus  that  of  the  third  equals  4  ;  the  sum 
of  3  times  the  reciprocal  of  the  first  and  twice  the  reciprocal  of 
the  second  is  less  by  1  than  5  times  the  reciprocal  of  the  third. 
Find  the  three  numbers. 

23.  In  a  certain  two-digit  number  which  equals  8  times  the  sum 
of  its  digits,  the  tens'  digit  exceeds  3  times  the  units'  digit  by  1. 
Find  the  number. 

24.  The  sum  of  the  digits  of  a  two-digit  number  is  12,  and  if 
the  digits  are  interchanged,  the  number  thus  formed  will  lack  12 
of  being  twice  the  original  number.     What  is  the  number  ? 

25.  The  sum  of  the  digits  of  a  3-digit  number  is  11 ;  the  double 
of  the  second  digit  exceeds  the  sum  of  the  first  and  third  by  1, 
and  if  the  first  and  second  digits  are  interchanged,  the  number 
will  be  diminished  by  90.     What  is  the  number  ? 

26.  The  third  digit  of  a  3-digit  number  is  as  much  larger  than 
the  second  as  the  second  is  larger  than  the  first ;  if  the  number 
is  divided  by  the  sum  of  its  digits,  the  quotient  is  15 ;  and  the 
number  will  be  increased  by  396  if  the  order  of  its  digits  is 
reversed.     What  is  the  number  ? 

27.  A  capitalist  invested  ^4000,  part  at  5%,  part  at  4%,  and 
found  that  his  annual  income  from  this  investment  was  ^175. 
How  much  was  invested  at  5  %,  and  how  much  at  4  %  ? 

Solve  this  problem  also  by  using  only  one  unknown  letter. 

28.  A  capitalist  invested  A  dollars,  part  at  p  %,  part  at  q%, 
and  found  that  his  annual  income  from  this  investment  was  B 
dollars.     How  much  was  invested  at  p  %  ?  at  g  %  ? 

Show  that  this  problem  includes  Prob.  27  as  a  special  case. 

29.  Divide  the  number  N  into  two  such  parts  that  1/m  of  the 
first  part,  plus  l/^i  of  the  second,  shall  exceed  the  first  part  by  M. 

Specialize  this  problem,  and  find  the  solution  of  the  special 
problem  by  substituting  in  the  general  solution. 

30.  Three  cities.  A,  B,  and  C,  are  situa,ted  at  the  vertices  of  a 
triangle ;  the  distance  from  A  to  C  by  way  of  B  is  50  miles,  from 
A  to  B  by  way  of  C  is  70  miles,  and  from  B  to  C  by  way  of  A  is 
60  miles.     How  far  apart  are  these  cities  ?     (Make  diagram.) 


106]  SIMULTANEOUS  SIMPLE  EQUATIONS  163 

31.  In  the  triangle  ABC,  AB  =  12  inches,  BC  =  10 
inches,  BE^BF,  FC=GG,  AG  =  4.iBF.  If  the 
perimeter  of  the  triangle  is  42,  find  AG,  AE,  BE,  FC. 

32.  A  quantity  of  water  which  is  just  sufficient  to 
fill  three  jars  of  different  sizes,  will  fill  the  smallest 
jar  exactly  4  times ;  or  the  largest  jar  twice,  with  4 
gallons  to  spare;  or  the  second  jar  3  times,  with  2 
gallons  to  spare.     Find  the  capacity  of  each  jar. 

33.  Two  men,  A  and  B,  rowed  a  certain  distance, 
alternating  in  the  work  ;  A  rowed  at  a  rate  sufficient  to  cover  the 
entire  distance  in  10  hours,  while  B's  rate  would  require  14.  If 
the  journey  was  completed  in  12  hours,  how  long  did  each  row  ? 

34.  Two  boys,  A  and  B,  run  a  race  of  400  yards,  A  giving  B  a 
start  of  20  seconds  and  winning  by  50  yards.  On  running  this 
race  again.  A,  giving  B  a  start  of  125  yards,  wins  by  5  seconds. 
What  is  the  speed  of  each  ?     Generalize  this  problem. 

35.  If  A  and  B  can  do  a  certain  piece  of  work  in  10  days,  A 
and  C  in  8  days,  and  B  and  C  in  12  days,  how  long  will  it  take 
each  to  do  the  work  alone  ? 

36.  A  and  B  together  can  build  a  wall  in  5^^  days;  being 
unable  to  work  at  the  same  time,  A  works  5  days,  then  B  takes  up 
the  work,  finishing  it  in  6  days  more.  In  how  many  days  could 
each  have  built  the  wall  alone  ?     Generalize  this  problem. 

37.  A  man  can  row  m  miles  downstream  in  c  hours  and  m 
miles  upstream  in  d  hours;  what  is  his  rate  of  rowing  in  still 
water,  and  what  is  the  rate  of  the  current  ? 

38.  From  the  solution  of  Prob.  37  find  the  solution  of  the 
special  problem  in  which  m  =  6,  c  =  1^,  d  =  4. 

39.  Two  trains  whose  respective  lengths  are  1200  feet  and  960 
feet  run  on  parallel  tracks;  when  moving  in  opposite  directions, 
the  trains  pass  each  other  in  24  seconds ;  when  moving  in  the 
same  direction,  each  at  the  same  rate  as  before,  the  faster  passes 
the  slower  in  1^  minutes.     Find  the  rate  of  each  train. 


164  HIGH  SCHOOL  ALGEBRA  [Ch.  X 

40.  Two  trains  are  scheduled  to  leave  the  cities  A  and  B,  m 
miles  apart,  at  the  same  time,  and  to  meet  in  h  hours ;  but,  the 
train  from  A  being  a  hours  late  in  starting,  and  running  at  its 
regular  rate,  the  trains  met  k  hours  later  than  the  scheduled  time. 
What  is  the  rate  at  which  each  train  runs  ? 

41.  From  the  solution  of  Prob.  40  iind  the  sohition  of  the 
special  problem  in  which  m  =  800,  /i  =  10,  a  =  If ,  and  k  =  Jq. 

42.  A  train  was  scheduled  to  make  a  certain  run  at  a  uniform 
speed.  After  traveling  2  hours  it  was  delayed  1  hour  by  an 
accident,  after  which  it  proceeded  at  -y-  its  usual  rate  and  arrived 
^  hour  late.  Had  the  accident  occurred  36  miles  farther  on,  the 
train  would  have  been  36  minutes  late.  Find  the  usual  rate  of 
the  train  and  the  entire  distance  traveled. 

43.  Two  boats  which  are  d  miles  apart  will  meet  in  a  hours  if 
they  sail  toward  each  other,  and  the  second  will  overtake  the  first 
in  b  hours  if  they  sail  in  the  same  direction.  Find  the  respective 
rates  at  which  these  boats  sail.  Also  discuss  fully  your  solution, 
i.e.,  interpret  the  results  (cf.  Prob.  4,  p.  141). 

44.  Find  an  expression  of  the  form  ax^  -\-bx-\-c  whose  value  is 

6  when  x  =  2,  3  when  x=  —1,  and  10  when  a?  =  4. 

Hint.  4  a  +  2b  +  c  is  the  value  of  ax^  +  bx  -\-c  when  x  =  2  ;  therefore, 
4a+26  +  c  =  6,  etc. 

45.  Find  an  expression  of  the  form  ax^  +  &x  +  c  whose  value  is 

7  when  x  =  S,9  when  x=  —1,  and  17  when  x  =  —  5. 

46.  Find  an  expression  of  the  form  ax^  +  bx^  -\-  ex  -\-d  which 
equals  — 16  when  x=  —1,-4  when  x  =  l,  —  43  when  x  =  —  2,  and 
— 100  when  x=  —  3. 

47.  Of  three  alloys,  the  first  contains  35  parts  of  silver,  to  5 
of  copper,  to  4  of  tin ;  the  second,  28  parts  of  silver,  to  2  of 
copper,  to  3  of  tin ;  and  the  third,  25  parts  of  silver,  to  4  of  copper, 
to  4  of  tin.  How  many  ounces  of  each  of  these  alloys  melted 
together  will  form  600  oz.  of  an  alloy  consisting  of  8  parts  of 
silver,  to  1  of  copper,  to  1  of  tin  ? 


10(5-107]  SIMULTANEOUS   SIMPLE  EQUATIONS  1()5 

48.  If  Prob.  47  demanded  merely  that  the  alloy  should  contain 
8  parts  of  silver  to  1  of  copper  (without  specifying  the  amount  of 
tin),  how  many  ounces  of  each  of  the  given  alloys  would  then  be 
required  ?     Why  is  this  problem  indeterminate  (of.  §  107)  ? 

107.*    Determinate    and    indeterminate    systems    of    equations. 

As  we  have  already  seen,  a  system  containing  as  many  inde- 
pendent equations  as  unknown  numbers,  can  always  be  solved,  i.e., 
the  unknown  numbers  can  be  determined  (§§  101-106).  Such  a 
system  is,  therefore,  a  determinate  system. 

On  the  other  hand,  a  system  in  which  there  are  fewer  inde- 
])eudent  equations  than  unknown  numbers  is  an  indeterminate 
system.  It  is  easy  to  show  that  this  statement  —  already  seen 
to  be  true  in  the  case  of  a  single  equation  containing  two  un- 
known numbers  (§  99)  —  is  true  generally. 

Thus,  suppose  we  have  three  equations  containing  four  unknown 
numbers.  By  regarding  one  of  these  numbers  temporarily  as 
kuown,  we  can  solve  the  given  equations  for  the  other  three ;  i.e., 
we  can  express  any  three  of  the  four  unknown  numbers  in  terms  of 
the  fourth.  To  every  assigned  value,  therefore,  of  this  fourth  un- 
known number,  there  corresponds  a  set  of  values  of  the  other 
three  (cf.  §  99)  ;  hence  the  system  is  indeterminate. 

Again,  there  can  never  be  in  a  system  more  independent  equa- 
tions than  there  are  unknown  numbers. 

For,  if  that  were  possible,  suppose  there  are  three  independent 
equations,  viz., 

ax-{-hy  =  c,  (1) 

hx+jy^k,  (2) 

and  Zfl?  -f  my  =  n.  (3) 

containing  but  two  unknown  numbers,  x  and  y. 

On  solving  (1)  and  (2)  we  obtain 

cj  —  bk        -,         ch  —  ak 

X  =  -^ and  y  = , 

aj  —  bh  '       bh  —  aj 

and  on  substituting  these  values  for  x  and  y  in  (3),  we  obtain 


\aj  —  bh)  \bh  —  qjj 


*  This  article  may  be  omitted  till  the  subject  is  reviewed. 


166  HIGH  SCHOOL  ALGEBRA  ICu.  X 

i.e.,  the  known  numbers  of  these  equations  are  not  independent 
(n,  for  example,  is  expressed  in  terms  of  a,  b,  c,  Z,  etc.),  hence  the 
given  equations  are  themselves  not  independent. 


REVIEW  EXERCISE-CHAPTERS  VI-X 

Find  the  H.  C.  F.  and  also  the  L.  C.  M.  of: 

1.  6a^  +  13a;-5  and3a^  +  2a.'2-f  2a;-l. 

2.  12a^-29a;  +  14and8a.-2-30  +  6a;*-f-lla^  +  33a;. 

3.  Chanere      '^  ^^~ to  an  equal  fraction  whose  denomi- 

nator  is  24  a?  —  6  a^x^ ;  also  to  an  equal  fraction  whose  numerator 
is  1  —  10  ay  —  5  a  +  2  y. 

Simplify : 

go;'"  —  bx'^"^''-  1  +  x      l  +  o^ 

**     a^ftic-dV  *  „     1  +  aJ-     1+a^ 


l-hx^      1  +  a^ 


5. 


4a^ 
6xy-\-9y^  .  Sa^-27f 


r^«I-^)_yAp-l)  ^  4^ 


^4  _  ^  _  J  ,^.2  _|_  ^,  _j_  g  4  ic^  —  6  ict/ 

1^1  1 


10. 


(a-6)(6-c)      (c-6)(c-d)      {d-c)(b-a) 
k-1  1-k 


(]c -  l)(k  -m)(n-  Tc)      (I -k){n- k){k -p) 

11.  Why  may  a  term  be  transposed  from  one  member  of  an 
equation  to  the  other  by  merely  changing  its  sign  ? 

12.  When  are  equations  conditional  ?  identical  ?  integral  ? 
fractional  ?  literal  ?  numerical  ?  indeterminate  ?  Illustrate  each 
of  your  answers. 

Solve,  and  check  as  the  teacher  directs : 

13.  3a;2_5i»-12  =  0.  2 -\- x ^     19 

14.  6m2-13m  =  -6.  '  2-a;~     21 


107] 


SIMULTANEOUS   SIMPLE  EQUATIONS 


167 


16.  1---^ 


3^ 

2s 


3s 


X     a 


17 


=  a-\-b. 


19.  {x-2y+(x+oy=(x  +  7y. 

20.  7x  +  5fl-^^  =  a(x-a). 


3    2 
ic    a 
18.  6x'-\-7x^-20a^  =  0. 

y  —  4:     ?/  —  6  _  y  —  5  _  y_ 


21. 


+ 


8v 


2vH-5     2v-5     25-4^2 


22. 


2/-5     2/-7     2/-6     2/-8 

23.  (a?  — a)(a  — &  +  c)  =  (icH-a)(6  — a  +  c). 

24.  Show  that  while  2  is  a  root  of  the  integral  equation  which 

results  from  clearing  — ^  + — rr-  =  8  H of  fractions, 

x-^o      {x  +  5)(x-2)  x-2 

it  is  not  a  root  of  the  given  fractional  equation.     How  could  we 

avoid  introducing  this  extraneous  root  ? 

25.  Form   the  equations  whose  roots  are:   2,-9;     —  3|-,  4; 
I,  f ;     -2a,  -6a;     1,3,-7;     l-c,c-l. 

26.  When  are  two  equations  equivalent  ?  inconsistent  ?  simul- 
taneous ?  independent  ?     Illustrate  each  of  your  answers. 

27.  Explain  the  term  '^ elimination"  as  applied  to  simultaneous 
equations,  and  outline  three  methods  of  elimination. 

Solve  the  following  systems ;  check  as  the  teacher  directs : 


28. 


29. 


7-2a;^3 
5-37/  2' 
y  —  x  =  4:. 


2x- 


y 


=  4, 


31. 


32. 


30. 


32,  +  ^  =  9. 


1  +  1-1 
X     y     4 


33. 


1_ 

12 


5  ahx  -\-2y  = 

166, 

3  a5a.'  +  4  ?/  = 

18  6. 

a     6 

3a     66     3 

y  —  z     x-\-z 

1 

2           4 

'2' 

x—y      x—z 

—  0 

5            6 

'-'> 

?/  4-  2;  _  a)  4-  ?/ 
4           2 

-4. 

168  HIGH   SCHOOL   ALGEBRA  [Ch.  X 

34.  If  asi^ -\-hx-\-c  becomes  8,  22,  42,  respectively,  when  a; 
becomes  2,  3,  4,  what  will  it  become  when  x  becomes  —  \  ? 

35.  The  sum  of  two  numbers  is  5760,  and  their  difference  is  \ 
of  the  greater.     Find  the  numbers. 

36.  What  number  added  to  its  reciprocal  gives  5.2  ? 

37.  It  takes  2000  square  tiles  of  a  certain  size  to  pave  a  hall, 
or  3125  square  tiles  whose  dimensions  are  1  inch  less.  Find  the 
area  of  the  hall  floor.  How  many  solutions  has  the  equation  of 
this  problem  ?     How  many  has  the  problem  itself  ? 

38.  Divide  the  number  a  into  two  parts  such  that  the  second 
part  shall  equal  n  increased  by  m  times  the  first  part. 

39.  What  number  must  be  added  to  m  and  to  n  in  order  that 
the  first  sum  divided  by  the  second  shall  equal  p/q  ?  What  does 
your  answer  become  when  p  =  q"}  What  does  this  indicate 
(1)  when  m  =  n,  (2)  when  m  and  n  are  unequal  ? 

40.  In  order  to  build  a  new  clubhouse,  a  country  club  assessed 
each  of  its  200  members  a  certain  sum  ;  later  an  increase  of  50 
in  the  membership  reduced  the  individual  assessments  by  ^10. 
Find  the  cost  of  the  proposed  house. 

41.  At  what  time  between  3  and  4  o'clock  is  the  minute-hand 
25  minute  spaces  ahead  of  the  hour-hand  ? 

42.  The  freezing  point  of  Avater  is  marked  0°  on  a  Centigrade 
thermometer,  and  32°  above  zero  on  a  Fahrenheit  thermometer. 
If  100°  Centigrade  =  180°  Fahrenheit,  find  the  reading  on  a  Centi- 
grade thermometer  corresponding  to  68°  Fahrenheit.  (Make  a 
diagram  of  each  scale.) 

43.  State  and  solve  the  general  problem  of  which  Prob.  42  is 
a  particular  case.  By  substitution  in  the  formula  thus  obtained 
express  in  the  Centigrade  scale  the  following  Fahrenheit  readings  : 
44°;  212°;    -10°;   0°. 

44.  A  man  rows  a  boat  with  the  tide  8  miles  in  If  hr.  and 
returns  against  a  tide  1  as  strong  in  4  hr.  What  is  the  rate  of 
the  stronger  tide  ?    At  what  rate  does  the  man  row  in  still  water  ? 


107]  REVIE]r   EXEliCISE  169 

45.  A  man  selling  eggs  to  a  grocer  counted  them  out  of  his 
basket  4  at  a  time  and  had  1  e^^  left  over ;  the  grocer  counted 
them  into  his  box  5  at  a  time  and  there  were  3  left  over.  If  the 
man  had  between  6  and  7  dozen  eggs,  how  many  must  there  have 
been  (cf.  §  99)  ? 

46.  Of  two  wheelmen,  A  and  B,  A  starts  c  hours  in  advance 
of  B,  and  travels  at  the  rate  of  a  miles  in  h  hours,  while  B  follows 
at  the  rate  of  p  miles  in  q  hours.  How  far  will  A  travel  before  he 
is  overtaken  by  B  ? 

Under  what  conditions  is  this  solution  positive  ?  negative  ? 
zero  ?  infinite  ?     Interpret  the  result  in  each  case. 


CHAPTER  XI 

INVOLUTION  AND   EVOLUTION 

I.    INVOLUTION 

108.  Introductory.  For  the  meaning  of  the  words  hase^ 
exponent^  and  power ^  as  used  in  algebra,  see  §§9,  30,  and  36. 
The  process  of  raising  a  number  or  expression  to  any  given 
power  is  called  involution. 

In  this  chapter,  as  in  the  earlier  treatment  of  powers, 
we  shall  use  only  positive  integers  as  exponents.  Later  on 
(Chapter  XVI),  however,  we  shall  find  it  advantageous  to 
employ  such  symbols  as  aP^  a~^^  and  or  also,  and  we  shall 
then  assign  suitable  meanings  to  such  symbols. 

109.  Even  powers,  odd  powers,  powers  of  fractions,  etc.     A 

power  of  any  given  number  is  called  even  or  odd  according 
as  its  exponent  is  even  or  odd. 

From  the  law  of  signs  given  in  §  18  it  follows  that : 

(1)  All  integral  powers  of  a  positive  number  are  positive. 

(2)  All  even  integral  powers  of  a  negative  number  are 
positive. 

(3)  All  odd  integral  powers  of  a  negative  number  are 
negative. 

And  from  §  83  (i)  [cf.  also  Ex.  30,  p.  121],  it  follows  that 


a^      fmY      w*      . 
J5'    [nJ^V''- 


Let  pupils  fully  explain  each  of  the  above  statements: 

170 


108-110]  INVOLUTION  AND  EVOLUTION  171 

EXERCISE  LXXIV 

1.  Answer  again  questions  18-20  on  p.  39. 

2.  Write  that  power  whose  base  is  k  and  whose  exponent  is 
m  — 3.  Are  there  any  limitations  here  on  the  value  of  A;?  on 
the  value  of  m? 

3.  From  the  definition  of  an  exponent  show  that  a^  •  ar^  =  a^. 
Also  that  2^.  2.  22  =  2^ 

4.  For  what  values  of  n  between  1  and  10  is  (—3)"  •  (— S)^" 
positive  ?     Explain. 

5.  Show  that  an  even  power  of  a  negative  number  is  positive. 

6.  How  is  a  fraction  raised  to  a  power  (of.  Ex.  30,  p.  121)  ? 
Illustrate  your  answer. 

Simplify  each  of  the  following  expressions : 


8. 


(-!!)■■  -  (-drj 


110.  Exponent  laws.  The  following  formulas  state  what 
are  known  as  the  exponent  laws.  The  bases  (<2,  5,  and  c)  stand 
for  any  numbers  or  algebraic  expressions  whatever,  but  the 
exponents  are  positive  integers. 

(i)  First  exponent  law.     a""  -  a""  =  a"*+«.  [§  30 

For,  just  as  a^  •  a^  =  (a  •  a  •  a)  •  (a  •  a) 

=  a^    i.e.,  «^+2. 
so,  too, 

a^  '  a"  =:  (^a  '  a  '  a  '"  to  m  factors)(a  -  a  •  a  •••  to  n  factors) 
=  a  '  a  •  a  '••  to  (m  +  n)  factors 

Similarly,  a"^  -  a""  -  a^  =  o^^^'-^p. 

HIGH  SCU.   ALG. — 12 


172  BIGH  SCHOOL  ALGKhRA  [Cn.  XI 

(ii)   Second  exponent  law.      (a'"y  —  a"***. 
For,  just  as         {aF)'^  =  {a  -  a  ■  a^ 

=  (^a  '  a  '  a^  '  (^a  '  a  '  a) 

=  a^,    i.e.^  a^'2; 
so,  too,  (^a'^y  =  (a  -  a  '  a  ••'  to  m  factors)** 

=  a  •  a  •  a  •  •  •  to  mn  factors 

(iii)   Third  exponent  law.     a^  -  b^  =  (aby. 
For,  just  as        a^  •  b^  =  a  •  a  ■  a  -  b  -  b  -h 
=  ab  •  ab  '  ab 
^(aby; 
so,  too, 

a"b^  =  (^a  •  a  •  a  ••'  to  n  factors)  -  (b  -  b  •  b  -"  to  n  factors) 
—  ab  '  ab  •  ab  "•  to  n  factors 

=  iaby. 

Similarly,  a^^c""  =  (obey. 

(iv)  Fourth  exponent  law.     a'"  -r-  a"  =  a""-".  [§  80 

This  law  is  an  immediate  consequence  of  (i)  above,  and 
of  the  definition  of  division  (§  8),  for  since 

therefore  a"»  ^  a"  =  «"*"". 

EXERCISE  LXXV 

Simplify,  and  explain  your  work  in  each  case : 
7.  (-5a)2. 

9.    (IT^S^)^ 

^    \2 

11.  '       ^ 


1. 

a'h^ '  ah\ 

2. 

3A-(- 

2a?f). 

3. 

a  '  0?  '  gC 

■a'. 

4. 

5. 

ccPe 

6. 

{x'zy. 

m- 


12. 

X'^  '  oc^. 

13. 

X""  -T-Olf. 

14. 

(x-y. 

15. 

(2x'"'y, 

16. 

s«  .  s^. 

17. 

V^  '  V^  '  v^. 

18. 

c^r 

110-111] 

INVOLUTION  AND  EVOLUTION 

-  (-ir'    -  m. 

-  C-^J- 

-  (5T-      -  c^; 

21.  (H^)*. 

22.  (-c)2- 

25.  (,,,•.->)'.                     ^^    r.„_ 

26.  (5  »•")».                          •  H^a  + 

173 


Write  the  following  as  powers  of  products  [cf.  law  (iii)  above]  : 

30.  /i%2.  33.   o?if.  36.   a'  •  (2  6)»^. 

31.  r^s¥.  34.   a^2/*-  37.   3*  •  (  -  m)^  •  (?i/. 

32.  c^d^  35.    —  2»  .  3^  38.    ir-"  •  /«  •  2^". 

39.  What  does  a  represent  in  the  proofs  of  §  110  ?    May  it  rep- 
resent a  polynomial  as  well  as  a  number  ? 

40.  Translate  the  first  exponent  law  (§  110)  into  verbal  lan- 
guage (cf.  §  30). 

41.  Translate  the  second,  third,  and  fourth  exponent  laws  into 
verbal  language. 

42.  Is  (a  •  6  •  cf  equal  to  a^  •  6^  •  c^  ?    Is  (a  +  6  -f  c  -h  df  equal  to 
a'-  -f  ^^  +  c^  +  c^^  ?     Explain  your  answers. 

43.  Is  [(-2)3]2  equal  to  [(-2)^^?     Why?     Is  (a^)*  equal  to 
{xy^     Why? 

111.   Powers  of  binomials.     We  have  already  seen  (§§  52 
and  57)  that 

and  (a  +  hy  =  a^-\-^a%^2>ah'^  +  h^. 

These  powers  (expansions)  were  obtained  by  direct  multi- 
plication, and  the  higher  powers  may,  of  course,  be  obtained 
in  the  same  way.     Thus, 
(a  +  6)4  =  ^4  4.  4  ^35  ^  6  ^252  +  4  aJ3  4.  54^ 
(a  -f  6)5  =  ^5  +  5  a^i  +  10  ^352  4. 10  aW  4-  5  aft*  _|_  js^ 
(a  4-  6)6  =  ^6  4.  (3  ^56  4. 15  ^4^2+  20  a%^+ 15  ^254 4. 5  ^554.  je,  etc. 


174  HIGH  SCHOOL  ALGEBRA  [Ch.  XI 

The  following  questions  may  serve  to  bring  out  the  strik- 
ing similarity  of  these  expansions : 

1.  How  does  the  exponent  of  the  first  term  in  each  ex- 
pansion compare  with  that  of  the  corresponding  binomial  ? 

2.  How,  in  each  expansion,  does  the  exponent  of  a  change 
as  we  pass  from  term  to  term  toward  the  right  ? 

3.  In  which  term  of  each  expansion  does  b  first  appear  ? 
How  does  the  exponent  of  b  change  from  term  to  term  ? 

4.  How  many  terms  in  each  expansion?  What  is  the 
sign  of  each  term  ? 

5.  What  coefficient  has  the  first  term  of  each  expansion  ? 
the  second  term  ? 

6.  Multiply  the  coefficient  of  any  term  in  any  of  the  ex- 
pansions by  the  exponent  of  a  in  that  term,  and  divide  this 
product  by  the  number  of  the  term ;  how  does  this  quotient 
compare  with  the  coefficient  of  the  next  term  ? 

7.  Assuming  that  the  expansion  of  (a  +  by  is  similar  in 
form  to  the  expansion  of  (a  +  ^)^  (a+i)^  etc.,  complete 
the  statement : 

(a  +  5)8  =  a8  +  8  a^b  +  28  a%^  +  .... 

112.  Binomial  theorem,  (i)  The  answers  to  the  first  six 
questioHS  in  §  111,  when  combined,  may  be  expressed  sym- 
bolically thus : 

(a  +  by  =  a--\-'^  a^-'b  +^(f -^)a"-2^,2 
^         "^  1  1.2 

1  .  ^  .  o 

This  formula,  which  was  discovered  by  the  celebrated 
English  mathematician  Sir  Isaac  Newton  (1642-1727),  is 
called  the  binomial  theorem;  its  correctness  is  proved  in 
§§  206-207. 


111-112]  INVOLUTION  AND  EVOLUTION  175 

(ii)  Since         a  —  5  =  aH-(—  5), 
therefore 
(a_5)3=[a+(-*)]3=«H3a2(-5)  +  3a(-6)2  +  (-6)3' 

?'.e.,  (a  —  by  differs  from  (a  +  b^  only  in  having  the  signs  of 
its  even  terms  negative.     So  also  for  other  powers  of  a  —  b,  ^.e., 

(a  -by=a^-^  a^-'b  +  ^i(!L:^^«-2j2 

1.2.3  «     ^  +      • 

EXERCISE   LXXVI 

Write  down  the  expansions  of  the  following  binomials : 

1.  (a  +  xy.  5.    {a-\-cy.  9.    (m2  +  6)3. 

2.  (mH-f)^  6.    (i»  +  2/)*.  10.    {m-\-hy. 

3.  (1*  +  ?;)^  7.    (2/  +  ;^)'.  11.    (m^+fty. 

4.  (p  +  g)'.  8.    {k-\-iy.  12.    (m2  +  63)6^ 

13.  Expand  each  of  the  following  expressions  :  (x-\-yy,  (^+2/)^ 
and  (x  +  yy ;  then  multiply  the  first  two  expanded  forms  together 
and  thus  verify  that  (x -\- yY  •  {x -\-  yy  =  (a;  +  yy. 

14.  What  terms  in  the  expansion  of  {c  —  dy  are  negative? 
Why? 

15.  Write  the  first  five  terms  of  (s  —  2  ty  and  simplify  your 
result  (cf.  Ex.  3,  p.  71). 

16.  How  many  terms  are  there  in  the  expansion  of  (m-f-?iy? 
How  many  in  (a  —  by  ?     How  many  in  (3  s  —  2  ty  ? 

Write  each  of  the  following  expressions  in  its  expanded  form  : 

17.  (k-cy.  23.  (4c  +  ^')'.  29.  {v'-2y. 

18.  (r  —  sy.  24.  (mii.  —  rsy.  30.  (2  xy  —  ly. 

19.  (m  —  ny.  25.  (ab  +  cdy.  31.  (c  +  a)». 

20.  (c4-dy«.  26.  (3a2  +  c«(^)^  32.  (2  771+3)1 

21.  (x'  +  yy.  27.  (A;4-l)'.  33.  (2 -3  0-2^)5. 

22.  (2r-ay.  28.  (^^-2)^  34.  (2-a'by. 


176  HiGn  SCHOOL  algebea  [Ch.  xi 

35.    (.r-.9vy.  gg_    fr-'lW  40-    [.ci  +  (b  +  c)J. 


36.  (2xy^-\-x'yy.  V        ^^  41.    (c  +  d  +  e)^ 

37.  fa-j-fl]  .  39.      -  +  -     •  ^  ^   ^ 

V       a?;  V«      V  43.    (2a^-m-l)^ 

44.  Write  the  first  four  terms  of  (a  +  x)'-^ ;  the  first  three  terms 
of  (a;  —  y)^ ;  the  first  three  terms  of  (2  ax  — 3  k^y. 

II.   EVOLUTION 

113.  Definitions.  Here,  as  in  arithmetic,  by  the  square 
root  of  any  given  number  we  mean  a  number  whose  square 
equals  the  given  number. 

Thus,  since  T^  =  49,  therefore  7  is  a  square  root  of  49. 

Similarly,  the  third  or  cube  root  of  a  number  is  a  number 
whose  third  power  equals  tlie  given  number. 

Roots  are  usually  indicated  by  the  radical  sign  ( V)i  which 
is  a  modification  of  the  letter  r,  the  initial  letter  of  the  Latin 
word  radix,  meaning  root.  A  small  figure,  called  the  index 
of  the  root,  is  written  in  the  opening  of  the  radical  sign  to 
indicate  the  particular  root  to  be  extracted.  When  no  index 
is  written,  the  index  is  understood  to  be  2. 

U.g.,  -yja  indicates  the  second  or  square  root  of  a,  (1) 

-^a  indicates  the  third  or  cube  root  of  a,  (2) 

and         -y/a  indicates  the  seventh  root  of  a,  (3) 

and,  in  general,  ya  means  the  rith  root  of  a, 

t.e.,  i^ay  =  a.  (4) 

An  indicated  root  is  said  to  be  an  even  root  or  an  odd  root 
according  as  its  index  is  an  even  or  an  odd  number. 

The  process  of  finding  a  root  of  any  given  number  is  called 
evolution  ;  it  is  the  inverse  of  involution  (cf.  §  108). 

Note.  In  practice  the  radical  sign  is  usually  combined  with  a  vinculum 
(§  11)  to  indicate  clearly  just  how  much  of  the  expression  following  the  radi- 
cal sign  is  to  be  affected  by  that  sign  ;  thus  \/9  +  16  means  tlie  square  root 
of  the  sum  of  9  and  16,  while  VO  +  16  indicates  that  16  is  to  be  added  to  the 
square  root  of  9. 


112-115]  INVOLUTION  AND  EVOLUTION  177 

114.  Law  of  signs  of  roots.  From  the  definition  of  root 
(§  113),  and  from  §  109,  it  follows  that  : 

1.  An  odd  root  of  any  number  has  the  same  sign  as 
the  number  itself.  Thus,  V8  =  2,  and  V—  8  =  —  2,  because 
23=8  and  (-2)3=  -8. 

2.  An  even  root  of  a  positive  number  has  two  opposite 
values,  i.e.^  one  positive,  the  other  negative.  Th us,  VSl  =  +  3 
or— 3,  since  (+ 3)*  =  (— 3)*=  81.  Instead  of  writing 
V81  =  -f  3  or  —  3,  we  usually  write  V81  =  ±  3 ;  this  expres- 
sion is  read,  ''The  fourth  root  of  81  equals  plus  or  minus  3." 

3.  An  even  root  of  a  negative  number  is  neither  a  positive 
nor  a  negative  number.  Thus,  V—  9  is  neither  +  3  nor  —3, 
since  ( -f  3)2  =  (  -  3  )2  =  +  9,  and  not  -  9. 

Note.  Such  indicated  roots  as  V—  9  are  called  imaginary  numbers 
(cf.  §§  146,  164)  ;  all  other  numbers  are,  for  distinction,  called  real  numbers. 
To  provide  for  such  roots  as  V— 9  we  must  again  extend  the  number  system, 
just  as  we  did  when  subtractions  like  3  —  8  first  presented  themselves  (cf . 
Chap.  II). 

115.  Roots  of  monomials.  If  a  monomial  is  an  exact 
power,  the  corresponding  root  can  usually  be  written  down 
by  inspection. 


E.g.y    </S  aV  =  2  a%  because  (2  a^xf  =  8  aV  (§  110)  ; 

■V'9^\>f=±3xy,  because  (+3  ay'y'y=(-Sxyy=9xY', 
i/-32x'^  =  -  2  x%  because  (-2x'y  =  -32  x''; 

^  =  2^^  because f2^Y  =  8™\ 

EXERCISE  LXXVII 

1.  What  is  meant  by  the  square  root  of  a  number  ?  Are  both 
5  and  —  5  square  roots  of  25  ?     Why  ? 

2.  What  are  the  square  roots  of  64  ?  the  fourth  roots  of  16  ? 
Why  ?  If  a  is  any  ecen  root  of  a  number,  then  —  a  also  is  a  root 
(with  the  same  index)  of  that  number,  —  explain, 


178  HIGH  SCHOOL   ALGEBRA  [Ch.  XI 

3.  What  is  the  cube  root  of  27  ?  of  -  27  ?  of  64  ?  of  -64  ? 
Explain.     How  does  -^32  compare  with  V—  32  ? 

4.  How  does  the  sign  of  an  odd  root  of  a  number  compare  with 
the  sign  of  the  number  itself  ?  Why  ?  Answer  these  questions 
for  an  even  root  also. 

5.  Give  the  cube  of  each  integer  between  1  and  7.  Name  the 
cube  root  of :    -  8 ;  1000 ;   -  1728 ;    -  f  7 .  ii5 .    _  216 ;  8000. 

6.  What  is  the  sign  of  any  even  power  of  a  positive  or  negative 
number?  Can,  then,  an  even  root  of  a  negative  number  be 
positive  ?    negative  ?     Illustrate  your  answer. 

7.  Is  — 13  a  square  root  of  169  ?  Why  ?  Is  5  as^  the  cube 
root  of  125  aV  ?  Why  ?  How  can  you  tell  whether  one  given 
number  is  a  square  root  of  another  given  number?  a  fifth  root? 

8.  How  do  we  find  the  exponents  in  the  cube  root  of  8  o}^y?'}f  ? 
in  the  4th  root  of  a%^h'  ?  in  the  6th  root  of  m^^^^  ?  in  the  nth 
root  of  a^^n^in  9     Explain. 

Find  the  following  indicated  roots,  and  check  your  answers. 
Also,  tell  which  are  even  and  which  are  odd  roots,  and  name  the 
index  in  each  case : 


9.    Va'6V^  16.    V128a^^6i^.  3/  21^  &d}'' 

22. 


343  (c-df 


10.  Vl6aV2/^           17  3       125<i/«,                           

•  A/      112%  ah^  23    xl^m^hK 

11.  ^32^«.  r^ -i,  ^225 

12.  V-243aiV.          ■  ^     128  0^^^     '  24. 


1000  c^ 
125  d'' 


13.  A/a^a^Y".  19.    V  J^  ^''^'"-  25     W^''""'"^'" 

14.  ^'/-64^.  20     A^/^^^y.  ^,       x|aF5^^ 


2V 


15. 


5/--32_aV^  3/.O27  gV  4/8I  (g-c)^^"^ 

\    2432/2^    '  '    \. 064  6V'  *    \    625  a*'"" » 

28.  Write  a  rule  for  the  extraction  of  such  roots  as  the  above, 
emphasizing  particularly  the  matter  of  exponents  and  signs. 
Does  your  rule  apply  to  roots  of  polynomials  also  ? 


116-116]  INVOLUTION  AND  EVOLUTION  179 

29.  Is  V9T16  equal  to  V9  •  Vi6 ?  Why  ?  Is  V9Tl6  equal 
to  V9  +  vTe  ?  Is  VoM^  equal  to  Vo^^-  Vfe'  (cf.  Ex.  42,  p.  173)? 
Is  Va^  •  h'^  equal  to  Va^  •  V6^  ?  State  in  words  your  conclusion 
as  to  the  square  roots  of  sums  and  products. 

116.  Roots  of  polynomials  extracted  by  inspection.     If  a 

polynomial  is  an  exact  power  of  a  binomial,  or  the  square  of 
a  polynomial,  a  little  study  usually  reveals  the  corresponding 
root. 

Ex.  1.   Find  the  square  root  of  m^  +  4  m^n  +  4  n^. 

Solution.  This  expression  is  easily  seen  to  be  (m^H-2n)^; 
therefore  Vm*^  +  4  m^n  -\-  4  n^  =  ±  (m^  +  2  n). 

Ex.  2.   Find  the  cube  root  of  8  a^  -  36  a^b  -27b^  +  54  ab^. 

Solution.  This  polynomial  consisting  of  four  terms,  two  of 
which,  viz.,  8  a^  and  —  27  b^,  are  exact  cubes,  may  be  the  cube  of 
a  binomial  (§  57) ;  if  so,  that  binomial  must  be  2  a  —  3  6.    (Why  ?) 

On  cubing  2  a  — 3  b,  we  see  that 

■\/Sa^-36a'b-27b^  +  54:ab^  =  2a-Sb. 

Ex.  3.   Find  the  square  root  of  a^  +  6^  —  2  a6  —  4  6c  +  4  c^  +  4  ac. 

Solution.  This  polynomial  consisting  of  six  terms,  three  of 
which  are  exact  squares,  and  three  of  which  are  double  products, 
7nay  be  the  square  of  a  trinomial  whose  terms  are  the  square  roots 
of  the  square  terms  (§  56).  A  little  further  examination  shows 
that         Va^  +  62  _  2  a6  -  4  6c  +  4  c^  +  4  ac  =  ±  (a  -  6  -f  2  c). 

EXERCISE   LXXVIII 

By  inspection  find  the  following  roots,  and  check  results  : 


4.    V4  aj- +  12  a; 4- 9.  6.    V(m+^)^  — 4  (m  +  ?i) +4. 


5.    V25 2/^ -40 2/ +  16.        7.    -Va^ -{-2xy +  y^- 2 xz~2yz-\-z\ 


a    V 8  u^  — 12  u^v  —  v^-\-6  uv^. 
9.    Va;'*  —  4:  oc^y  -^  y^  —  4:  xy^  +  6  icy. 
10.    v^8  /r  -  84  h^k  +  294  hk^  -  343  Ar^. 


180  HIGH  SCHOOL  ALGEBRA  [Ch.  XI 

11.    VciP-  b'-5a'b-^5  ab*  + 10  a^b'  -  10  a%\ 


12.  Va'  +  962_6a6  +  6(a;-2i/)(a-36)4-9(a^-4a;y  +  4?/2). 

13.  ^x^  -6aba:^-\- 15  a^^x*  -  20  a'b^x^-{- 15  a'b'x^-  6  a^6'x  +  a'6^ 

117.   Square  roots  of  polynomials.* 

Sincev  (A;  +  w)^  =  /i:^  -f-  2  A:2*  +  w^^ 

therefore  Vk^  -^  2ku-tu^=  k -\-  u, 

and  we  shall  now  try  to  find  a  method  by  which  the  root, 
k  -\-  u,  may  be  found  from  the  power,  k^  +  2ku  +  u^. 

Manifestly  the  first  term  of  the  root  (viz.,  k)  is  the  square 
root  of  the  first  term  of  the  power. 

And  having  subtracted  k'^  (the  square  of  this  root  term) 
from  the  power,  the  next  term  of  the  root  (viz.,  u}  is  found 
by  dividing  the  first  term  of  the  remainder  (viz.,  2ku-\-u^) 
by  twice  the  root  term  previously  found  (viz.,  2  k^. 

The  actual  work  may  be  arranged  thus : 

k'^  +  2  ^•M  +  u^  \k  -\-u 

T^ 

Trial  divisor f       =2k 


Complete  divisor  =  2  A;  + 


2  ku  +  w2 

'Iku  +  u'^  =  (2  k  +  u)  u 


0 
The  same  method  may  be  applied  to  other  polynomials.  ^ 

Ex.  1.    Find  the  square  root  of  4  s-  -  28  .s^  +  49  f. 

Solution.  Let  k  represent  the  first  term  of  this  root,  and  u 
the  next  term  ;  then 

4  s^  -  28  st^  +  49  t'  contains  (k  +  uf,  i.e.,  k-  +  2  few  +  u\ 

Now  the  first  term  of  the  square  root  of  4  s^  —  28  st^  +  49  f 
is,  manifestly,  2s,  i.e.,  k  =  2s;  and  the  next  root  term  may  be 
found  as  above,  thus  : 

*  For  a  more  detailed  discussion  of  this  topic,  see  El.  Alg.  §  125. 

t  Twice  the  root  already  found  at  any  stage  of  the  work  is  usually  called 
the  trial  divisor  (T.  D.)  and  the  trial  divisor  plus  the  next  root  term  is  called 
the  complete  divisor  (C.  D.).  %  See  Note  1,  p.  181. 


10-117]  INVOLUTION  AND  EVOLUTION  181 

4  .s2  -  28  .<?«3  +  49  «« I  2  s  -  7  i-"^ 


k^  =  (2  s) 


•2-4  .V ' 


T.T>.=2k  =  4s 

C.  D.  =2^  +  w=4s-7<5 


-  2«.s'«'  +  49  «6 

-  28  st^  +  49  ^  =  (2  A:  +  m)  w 


Checks  :  (1)  Square  2  s  —  7  f,  or  (2)  substitute  special  values 
for  s  and  t  (gL  §  25). 

Ex.  2.    Find  the  square  root  of  9  x^  -\-  6  x^  —  11  x^  —  4:  x  -{-  4. 

Solution.  At  any  stage  of  the  process  of  finding  this  root,  let 
k  represent  the  term  or  terms  already  known,  and  let  ii  represent 
the  next  term  ;  then 

9x^-{-6x^  —  lla^  —  4:X-\-4:  contains  k^-{-2ku-\- \C\  [§  h^ 

Here  the  first  term  of  the  root  is  3  a^,  i.e.,  k  =  3x^,  and  the  next 
term  (u)  may  be  found  as  in  Ex.  1,  thus : 

9x*+6a;3-llx2-4x+4  I  3  y.2  +  X  -  2 
k^  =  (3  a;2)2  =  9x4 


T.  D.  =  2  ^•  =  6  a;2 

C.I).=  2k-{-  u  =  6x^  +  x 

T.B.  =  2k*  =  6x^  +  2x 

C. D.  =  2  A;  +  u  =  6x^  +  2X-2 


(5  x3  -  1 1  x2  -  4  X  +  4 

0  y3  +      .7:2 =    (2  A:  +  it)  u 


12x2 -4x  + 4 

12  x2  -  4  X  +  4  =  (2  k*-h  n)  u 


Checks :  (1)  Square  Sx'-{-x-2',  or  (2)  use  §  25. 

Note  1.  Before  applying  the  process  of  Exs,  1  and  2  a  polynomial  should 
be  arranged  according  to  ascending  or  descending  powers  of  one  of  its 
letters. 

Note  2.  Exs.  1  and  2  show  how  to  find  the  square  root  of  a  polynomial 
which  is  an  exact  square  ;  i.e.,  if  the  above  process  is  continued  until  a  zero 
remainder  is  reached,  then  the  square  of  the  root  thus  found  will  be  the  given 
polynomial.  If,  however,  the  same  process  is  applied  to  a  polynomial 
which  is  not  an  exact  square,  then  as  many  root  terras  as  desired  may  be 
found,  and  the  square  of  this  root,  at  any  stage  of  the  work,  equals  the 
result  of  subtracting  the  corresponding  remainder  from  the  given  polynomial ; 
such  a  root  is  called  an  approximate  root,  and  also  the  root  to  n  terms. 


*  Here  k  represents  3  x2  +  x,  and  u  represents  —  2.  Observe  also  that  the 
first  and  second  subtractions  in  this  solution  are  together  equivalent  to  the 
subtraction  of  (3  x2  +  x)2  from  the  given  expression. 


182  IIIGB  SCHOOL  ALGEBRA  [Ch.  XI 

EXERCISE  LXXIX 

Find  the  square  ropt  of  each  of  the  following  expressions,  and 
check  your  work : 

3.  9  m^*  -  66  m2  + 121.  5.  4: -\- S  x  -  4:  a^ -\- x\ 

4.  16/ +104 7-^ +  169.  6.   l  +  2m-3m2-4m3  +  4m^ 

7.  l-6y-{-5y^-\-12f-\-4:y\' 

8.  9a*  +  30a^a;  +  aV-40aa^  +  16a;*. 

9.  4a^  +  17a^-22.r^  +  13a;*-24a;-4a^  +  16. 

10.  4  a^  +  64  6*  -  20  a^b  +  57  a^b^  -  80  a¥. 

11.  6a^2/  +  2a;«i/^-28aj.v^  +  9.T«  +  42/«  +  45a^/  +  43icy. 

12.  3x''-2:f^-af-i-2x  +  l-\-x\ 

13.  48a*  +  12a2+l-4a-32a3  +  64a«-64a^ 

14.  46x'  +  25x'-Ua^-^0x  +  4:xr'-\-25-12xi', 

15.  x'^ -  2 x'^y  +  2  afz^  -2  yz^  +  y^ -hz\ 

16.  ^  +  16ay  +  8a^v'.  18.  9  a;2_24a;  + 28-^^ +  i. 

17.  aj2_j.2a;_i_?+l  :*        19.   4^2- 20« +  21 +  — +  i. 

X     x^  a      a^ 

20.  n^  +  4  71^  +  -i  +  2  7i  +  4  +  4  n^. 

21.  a)''  +  i  +  4aj3  +  i  +  6a^  +  -i-  +  5  +  5a;  +  5. 

cc*  ar  4  ar  x 

22. ^4_^_L..ii_. 

r       /  ^      4e^ 

23.   (a; - 2/)2 -2{xy -i-xz- y^ - yz)  +  (y  +  zf. 

25.  1  +  »,  to  three  terms  (cf.  Note  2,  p.  181). 

*  Observe  that  this  expression  is  already  arranged  according  to  descending 
powers  of  x. 


117-118]  INVOLUTION  AND  EVOLUTION  183 

26.  1+2  m-,  to  four  terms. 

'27.  a^  -f  1,  to  three  terms. 

28.  1  -f-.aj  — a^,  to  four  terms. 

29.  X* -\- 2  afy  -{- y* -\-  xy^  +  ar^2/^  ^o  four  terms. 

30.  In  Ex.  1  is  not  —  2  s,  as  well  as  -f  2  s,  a  square  root  of  the 
first  term  ?     Solve  Ex.  1,  using  —  2  s.     Does  your  result  check  ? 

31.  Solve  Ex.  2,  using  —Sx^  as  the  square  root  of  the  first 
term,  and  compare  your  answer  with  that  found  in  the  text. 

32.  By  extracting  the  square  root  until  a  numerical  remainder  is 
reached,  show  that  x*-\-4:a^-{-SQif-\-Sx—5  equals  (ic-+2a;  + 2)^—9, 
and  thus  find  the  factors  ofic'*  +  4a^  +  8a:^  +  8ic  —  5. 

33.  As  in  Ex.  31,  find  the  factors  of  x'*'-\-6af-\-llx^-\-6x  — S; 
also  of  a«  -  6  ci^  + 10  a^  +  9  a^  _  30  a  +  9. 

118.  Square  roots  of  arithmetical  numbers.*  In  order 
to  proceed  systematically,  and  find  the  successive  digits  of 
the  root  in  their  order  from  left  to  right,  we  first  separate 
the  given  number  into  periods  of  two  figures  each,  toward  the 
right  and  left  from  the  decimal  point.  The  root  may  then 
be  extracted  by  virtually  the  same  process  as  that  used  in 
§117. 

Note.  The  reason  for  the  separation  into  periods  lies  in  this :  the  square  of 
any  number  of  tens  ends  in  two  ciphers,  and  hence  the  first  two  digits  at  the 
left  of  the  decimal  point  are  useless  when  finding  the  tens'  digit  of  the  root ; 
they  are,  therefore,  set  aside  until  needed  to  find  the  units'  digit  of  the  root. 
So,  too,  the  square  of  any  number  of  hundreds  ends  in  four  ciphers,  and  hence, 
for  a  like  reason,  two  periods  are  set  aside  when  the  hundreds'  digit  of  the 
root  is  being  found,  and  so  on.  Similarly  for  the  periods  at  the  right  of  the 
decimal  point. 

Ex.  1.  Eind  the  square  root  of  1156. 

Solution.  This  number  consists  of  two  periods,  hence  its 
square  root  consists  of  two  digits.     Again,  since  9  is  the  greatest 

*  For  a  more  complete  discussion  of  this  topic  see  El.  Alg.  §  126. 


184 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XI 


square  in  the  left-hand  period,  therefore  3  is  the  first  figure  in  the 
root.  Now,  let  k  represent  the  known  part  of  the  root  at  any  stage 
of  the  work,  and  u  the  next  root  figure,  then 

1156  contains  Tc^  +  2ku-\-  u^, 

and  the  work  may  be  arranged  as  follows : 


k^  =  (30)'-^  = 
T.  D.  =  2  A:  =  60 
C.  D.  =  2  ^'  +  w  =  60  +  4 


11'56  I  30  +  4  : 
900 


34 


256 
256 


{2  k  +  u)u 


Check:    (34)2  =  1156. 
Ex.  2.   Find  the  square  root  of  315844. 
Solution.     Using  k  and  u  as  in  Ex.  1,  we  have 
315844  contains  A;^  +  2  ku  +  u% 
and  the  work  may  be  arranged  as  follows : 


fc-^  =(500)2  = 

31'58'44  1  500  +  60  +  2  = 
250000 

562 

T.D. 
CD. 

=  2  A  =          1000 
=  2k  +  u  =  1060 
=  2k*       =1120 
=  2k  +  u  =  1122 

65844 

63600  =(2  A;  +  u)u 

T.D. 
CD. 

2244 

22U  =(2  k-\-u)u 

Check:  (562)=-' =  315844. 

Note.  When  some  familiarity  with  the  above  process  has  been  gained, 
the  work  may  be  abridged  by  omitting  unnecessary  ciphers,  as  shown  below 
in  finding  the  square  root  of  315844  and  of  10.5625. 


31'58'44 
25 


10.'66'25 
9 


13.26 


106 
1122 


658 
636 


2244 
2244 


62 
645 


156 

124 
3225 
3225 


*  Here  k  =  560  ;  compare  footnote,  p.  181. 


118-119]  INVOLUTION  AND  ^VOLUTION  185 

EXERCISE   LXXX 

Extract  the  square  root  of  each  of  the  following  numbers,  and 
check  your  results : 

3.  1296.  6.   9216.  9.    667489.  12.    17424. 

4.  841.  7.    12.96.  10.   26.2144.  13.   36.8449. 

5.  2209.  8.    62.41.  11.   1664.64.  14.   101.0025. 

15.  How  may  the  square  root  of  a  fraction  be  found? 
Illustrate,  using  the  fractions  ^\  and  f||.  Is  — 14  also  a  square 
root  of  the  latter  fraction  ?     Why  ? 

16.  A  number  contains  one  decimal  place ;  how  many  decimal 
places  in  its  square?  How  many,  if  the  number  contains  two 
decimal  places  ?  if  it  contains  three  ?  if  it  contains  n  ?     Explain. 

17.  Show  from  Ex.  16  that  if  the  right-hand  period  of  a  decimal 
is  incomplete,  we  must  annex  a  cipher  to  complete  it.  Is  this 
true  of  the  left-hand  period  of  an  integral  number  also  ? 

18.  Extract  the  square  root  of  2  to  two  decimal  places  (cf. 
p.  181,  Note  2).  How  many  periods  of  ciphers  must  be  annexed  to 
2  for  this  purpose  ?     Why  ? 

Find  the  square  root  of  each  of  the  following  numbers,  correct 
to  two  decimal  places : 

19.  13.5.  21.    .017.  23.    |.  25.    4|. 

20.  |.  22.    1.1105.  24.    ■^.  26.    .049. 

27.  Is  V36  equal  to  V9-  Vi?  Is  V27  (i.e.,  V9T3)  equal  to 
3V3  where  the  roots  are  extracted  to  two  decimal  places?  to 
three  decimal  places  ? 


28.  Is  V450  (i.e.,  •\/225  •  2)  equal  to  15  V2  where  the  roots  are 
correct  to  two  decimal  places  ?  Show  that  V96  and  4  V6  are 
equal,  to  at  least  two  decimal  places. 

119.*  Cube  root  of  polynomials.  The  procedure  here  is  like 
that  in  §  117. 

*  Articles  119,  120,  with  Exercises  LXXXI  and  LXXXII,  may,  if  the 
teacher  prefers,  be  omitted  till  the  subject  is  reviewed. 


186  HIGir  SCHOOL  algebra  [Ch.  XI 

Since  Qc^uy  =  7c^-{-^k^u-h3ku^-hu\ 

therefore  VA;^  +  3  k'^u  -\-  3  ku^  -{-u^  =  k-{-  u. 

And  this  equation  shows : 

(1)  that  the  j^rs^  tei^m  of  the  cube  root  (viz.,  k)  is  the  cube  root 
of  the  first  term  of  the  polynomial ; 

(2)  that  the  trial  divisor  for  finding  the  next  term  of  the  root 
is3A^; 

(3)  that  the  complete  divisor  is  3  A;^  +  3  ku  -{-u^. 

The  actual  work  may  be  arranged  thus : 

F  +  3  khi  +  P>  ku^  +  v^  \k-\-u 

J^ 

T.  D.  =  3  k^ 


C.  D.  =  3  A:2  +  3  ku  +  u 


3  A:2m  +  3  kv!^  +  u^ 

3  k'^u  +  ^ku^  +  m3  =  (3  k'^  +  3  A;?(  +  ifi)  u 


0 


If  now  we  let  k  represent  the  part  of  the  root  already  known 
at  any  stage  of  the  work,  and  let  u  represent  the  next  term,  then 
the  above  method  will  serve  to  extract  the  cube  root  of  any 
"arranged"  polynomial  (cf.  §  117,  Exs.  1  and  2). 

Thus,  the  cube  root  of  ofi  -9x^  +  30 x^  -  i^x^  +  SOx'^  -  9x  +  1  may  be 
found  as  follows : 

I  a;2  -  3  a;  +  1 
5c6  _  9  a:^  +  30  0^4  -  45  a:3  +  30  ic2  _  9  X  +  1 

(x^y  =  x6  

T.  D.  =3(^2)2  =  3  X* 


CD.  =3x4-9x3  +  9^2 
T.D.  =3(x2-3x)2  =  3x*- 18x3 +  27x2 
C.  D.  =  3  (x2  -  3  x)2  +  3  (x2  -  3  X)  +  1 
=  3x*-  18x3  + 30x2 -9x  +  1 


-  9x5 +  30x4 -45x3  + 30x2 -9x  +  l 
-9x5  +  27x4-27x3 


3x4-  18x3  + 30x2 -9x  +  1 
3x4  -  18x3  + 30x2 -9x  +  l 


0 


EXERCISE  LXXXI 

Find  the  cube  root  in  Exs.  1-14,  and  check  your  results ; 

1.  Sa^-12x^  +  6x-l. 

2.  27x^-189x'y-hUlxy^-34.Sf/ 

3.  125  n""  -  150  mn^-Sm^-\- 60  m'n. 

4.  225uh  +  lS5uv^  +  125u''-\-27'i^. 


liy-120]  INVOLUTION  AND   EVOLUTION  187 

5.  a;^-20a^-6aj-f  15a;4-6x-^4-15ic2  +  l. 

6.  3aj^  +  9a;^-|-ic''  +  8  +  12a;  +  13a^  +  18a;l 

7.  342  x^  - 108  a;  -  109  a^  +  216  + 171  x""  -21x^  +  27  x\ 

8.  156a;*-144aj«-99a^  +  64aj6  4-39aj2-9a;  +  l. 

9.  u X ■\- - -112 -^  +  — +  0^-12 xUai.  Ex.  17,  p.  182). 

10.  20  +  i^  +  15c2  +  c«  +  |  +  i  +  6o^ 

11.  25  +  ^  +  82/«  +  302/-12  2/-^-25-i|. 

2/     2/  y 

12.  6  aV-4aV  -2  aV  +  6  aV  +  3  a^a;  +  «' +  a^^  -  3  a«8. 

13.  108  'ifz  -21  f-  90  2/V  +  8  2«  -  80  y V  ^  50  yh^  +  48  2/2^ 

14.  'y3»  -f-  9  ^Sn-S  ^  21  i;3"-2  —  42  'y3«-4_  35  -y3n-5_9  ^3«-l_g  ,y3«-6^ 


15.  Find  the  first  three  terms  of  v  1  +  x. 

16.  Find  the  first  four  terms  of  V 1  —  3  a?  +  aj^. 

120*  Cube  root  of  numbers.  The  cube  root  of  a  number  may  be 
found  by  virtually  the  same  process  as  that  used  in  §  119  for 
finding  the  cube  root  of  a  polynomial  (cf.  §§  118  and  117).  The 
number  should  be  separated  into  periods  of  3  figures  each,  begin- 
ning at  the  decimal  point  (why  ?),  and  the  right-hand  period,  if 
incomplete,  should  be  completed  by  annexing  ciphers. 

Ex.  1.   Find  the  cube  root  of  42875. 

SOLUTION 

k  -\-  u 
42'875  1 30  +  5  =  35 
27000 


15875 


T^  =  (30)8  = 
T.D.  =  3  A:2  =  3  .  (30)2  ^  2700 
CD.  =Sk^  +  'Sku  +  u^  , 

=  3  (30)2  +  3  (30)5  +  52  =  3175   1 15875 

0 


*  See  footnote,  p.  185. 

HIGH  SCH.  ALG. — 13  • 


188  HIGH  SCHOOL   ALGEliRA  [Cii.  Xi 

Ex  2.   Find  the  cube  root  of  9825.17,  correct  to  tenths. 

SOLUTION 

F  =  (20) 


9'825.'170  |20+  1  +  .4  =  21.4 
8000 


564.170 
539.344 


T.D.  =  3  k-^  =  3  (20)2  ^  1200        1825 
CD.  =3^•2  +  3^•«  +  M2 
=  3  (20)2  +  3  (20) .  1  +  12  =  1201        1261 
T.D.  =  3A;2=:  3(21)2  =1323 
CD.  =  3  (21)-^  +3  (21) (.4)  +  (.4)2  =  1348.36 

24.826 
Check.     (21.4)=^  =  9825.17  -  24.826  (cf.  Note  2,  p.  181). 

EXERCISE  LXXXII 
Extract  the  cube  root  of  each  of  the  following  numbers : 

1.  1728.  3.   31855.013.  5.   39304. 

2.  571787.  4.   148877.  6.   426.957777. 

7.  75.686967.  9.    .04,  to  two  decimal  places. 

8.  34.7,  to  two  decimal  places.     10.   3^,  to  two  decimal  places. 

121.  Transformation  of  indicated  roots.*  From  the  defini- 
tion of  the  symbol  -Va  (§  113)  it  follows  that,  whatever  the 
values  of  h  and  k, 

■Vm  =  hVk:  (1) 

for  (h-Vky^JiVk-h^k 

=  hh--Vk^ 

=  h^k,  [since  Vk  Vk  =  k~\ 

i.e.,  (hVky=h%, 

and  h  VJ  is,  therefore,  a  square  root  of  hj^k. 

Equation  (1),  read  forward,  tells  how  to  simplify  an  indi- 
cated square  root  of  a  number  which  contains  a  square  factor  ; 
and  read  backward,  it  tells  how  to  insert  a  coefficient  under 

*  Omit  §  121  if  radicals  are  to  be  studied  before  quadratics  :  see  Preface. 


120-121] 


INVOLUTION  AND  EVOLUTION 


189 


a  square  root  8i(/n.      Let   the   pupil  translate   this   eqiuttion 
into  verbal  language,  reading  it  both  ways. 

It  also  follows  from  the  definition  of  -Va  (§  113)  that 

(V^^0^=-^-  (2) 

Equations  (1)  and  (2)  will  be  useful  in  Chapter  XII. 

EXERCISE  LXXXIII 

Simplify  the  following  expressions  [cf.  Eq.  (1),  §  121]: 
1.  Vl2. 


2.  V50. 

3.  V48. 

4.  V63. 


5.  V4a26. 


'■4-  ■ 


"■  x/f- 


20. 


|V9. 


23.   —5cVb. 


6.  V54ajy. 
Insert  the  following  coefficients  under  the  radical  signs 

15.  3V2.  19.  -i-VS.  22.  aV2^. 

16.  5V7. 

17.  —  4V5. 

18.  7ViO.  21.  f Vii.  24.  |aV6'-4ac. 
Expand  the  following  expressions,  and  unite  like  terms : 

25.  (3+V5)l       28.  (6-V^2)l  31.  [^(3-2V5)?. 

26.  (3  +  V^^)'.   29.  (&-V6'-4ac)l 


27.  (-H-V3)'.  30.  (6-V4ac-62)^ 


32. 


D 


&  +  V62  -  4 


ac 


2a 


CHAPTER  XII* 

QUADRATIC  EQUATIONS  (Elementary) 

I.   EQUATIONS  IN  ONE  UNKNOWN  NUMBER 

122.  Definitions.  A  quadratic  equation  has  already  been 
defined  (§  93)  as  an  equation  wliicli,  when  simplified,  is  of 
the  second  degree  in  the  unknown  number  or  numbers. 
Thus,  3  §2  _  4  =  7  s^  aoiP'  +  5a;  +  c  =  0,  and  dm  =  4:m^  are 
quadratic  equations  in  s,  x^  and  7n,  respectively. 

By  transposing  and  uniting  terms  every  quadratic  in  x^ 
say,  may  evidently  be  reduced  to  the  standard  form 

ax^  -{-  bx  -^  c  =  0, 
wherein  a,  6,  and  c  represent  known  numbers,  and  are  usu- 
ally called  the  coefficients  of  the  equation.     The  term  free 
from  X,  viz.,  c,  is  called  the  absolute  (also  constant)  term. 

U.g.^  by  transposing,  etc.,  2x'^-^b  —  Sx  =  lx— S  becomes 
2:^2  —  10a;-|-13  =  0:  hence,  for  this  particular  equation  a=  2, 
5  =  -10,  and  c  =  U.  Similarly,  6x^- 2x=Sx^- 4:-2x 
becomes  3  a:^  -}-  4  =  0 ;   here  a  =  3,  5  =  0,  and  c  =  4:. 

An  equation  of  the  form  ax^  +  c  =  0  is  often  called  a  pure 
quadratic,  while  one  containing  both  the  first  and  second 
powers  of  the  unknown  number 'is  called  an  affected  quadratic. 

123.  Solution  of  pure  quadratics.  All  pure  quadratic  equa- 
tions may  be  solved  like  Exs.  1  and  2  below. 

Ex.  1.    Given  3  a;^  _  12  =  0 ;  to  find  x. 

*  Chapter  XII  may,  if  the  teacher  prefers,  be  omitted  until  Chapters  XIV 
and  XV  have  been  studied:  see  Preface. 

190 


122-123]  QUADBATIC  EQUATIONS  191 

Solution.   On  dividing   through   by  3,   and  transposing,   the 

given  equation  becomes 

a;2  =  4, 

whence  x=  ±2,* 

i.e.,  a;  =  2ora;=— 2; 

and  each  of  these  values  is  found  to  check. 

45  s^ 

Ex.  2.    Solve  the  equation  5 -— -  =  0. 

16 

Solution.     On  dividing  the  given  equation  through  by  5,  clear- 
ing of  fractions,  and  transposing,  we  obtain 

whence  3  s  =  ±  4, 

I.e.,  s=4ors  =  —  I; 

and  each  of  these  values  is  found  to  check. 


EXERCISE  LXXXIV 

Solve  and  check : 

3.  4a^  =  36.  11.   3cx'-10Sc^  =  0. 

4.  x'  =  ^.  12.   (a  +  l)V  =  4a2. 

5.  i_a^=_48.  13.    (k-6y  =  72-^12k. 

6     —  =  27  ■*■*•   <^-^)  =  2-^^' 

4  *  15.   a;(a;4-l)+3a^  =  x  +  |. 

7.  ^-Sy  =  0.  16.    (r-Sf  =  25. 

^y  17.  (u+iy-^%=o. 

8.  4('y2  +  3)=32.  18.    (a?-a)2  =  9&l 

9.  a^  +  3a;  =  3(a;+ll)-ll.        19.   4a^-l  =  a2  +  2a. 

^      (,  +  3)(r-3),  20.    ^(x-^)=(l-f 

9  8  16^  ^     V       8. 

*  Using  the  double  sign  in  each  member  here  gives  ±  x=  ±2, 
i.e.,  either  x  =  2,  (1)  or  -  a;  =  2,  (3) 

or  a;  =  -  2,      (2)  or  -  ic  =  -2.  (4) 

But  (3)  and  (4)  give  the  same  values  of  x  as  (2)  and  (1),  respectively; 
hence  in  solving  such  an  equation  as  x^  =  4,  the  double  sign  need  be  used  in 
one  member  only. 


192  HIGH  SCHOOL   ALGEBRA  [Ch.  XII 

21.  If  X  and  y  stand  for  unknown  numbers,  tell  which  of  the 
following  equations  are  simple,  and  which  quadratic  (cf.  §  93)  : 

aV -\-a^x-{-a  =  0\      ~    =-;  6x—ly  =  ll;  b  x  +  xy  —  1  y  — 11', 

Z         X 

y  2/  +  2 

22.  Reduce  5a:^H-2  —  8a;  =  4(8  —  x)  to  the  " standard  form." 
What  are  its  coefficients  ?     What  is  its  absolute  term  ? 

23.  Are  the  equations  in  Exs.  3-14  pure  or  affected?  Explain. 
What  is  the  absolute  term  in  Ex.  9? 

24.  Show  that  the  equation  of  Ex.  16  is  a  pure  quadratic  in 
r  —  3  but  an  affected  quadratic  in  r. 

Solve  each  of  the  following  equations  for  each  letter  it  contains  : 

25.   S  =  \qt\  26.     1  =  ^.        27.    E=^^       28.    R  =  K-- 

s       V-  2  d- 

A 

29.  The  square  on  the  hypotenuse  (longest  side)  of  a 
right-angled  triangle  equals  the  sum  of  the  squares  on 
the  other  two  sides.  In  the  right-angled  triangle  ABC, 
the  hypotenuse  AO  =  5,  while  -B(7=|  AB;  find  AB 
and  BC. 

30.  The  area  of  a  square  is  169  square  inches;  find  the  perime- 
ter and  the  diagonal  of  the  square. 

31.  How  many  rods  of  fence  will  inclose  a  square  garden 
whose  area  is  2^  acres  ? 

E  B  32.  From  a  rectangular  field,  ABCD,  whose  width 
is  f  of  its  length,  there  is  cut  off  a  square  field, 
AEFD,  whose  area  is  10  acres.  Find  the  area  of 
the  rectangular  field. 

33.    The   surface  area  of  a  cube  is  150  square 
inches.     Find  one  edge  and  also  the  volume  of  the  cube. 

124.  Quadratics  solved  by  factoring.  A  quadratic  equation 
ma}^  often  be  easily  solved  by  reducing  it  to  standard  form 
(§  122),  and  then  factoring  its  first  member  (§  72). 


123-124]  QUADRATIC  EQUATIONS  198 

Ex.  1.    Solve  the  equation  3  ic^  +  4  =  ar'  —  li  a;  +  16. 

Solution.     On  transposing,  uniting,  and  dividing  through  by 
2,  the  given  equation  becomes 

ar  -h  a;  —  6  =  0, 
i.e.,  (x-2){x-{-3)  =  0. 

Now,  as  in  §  72,  this  last  equation  is  satisfied  when 
a^-2  =  0  or  ic-|-3  =  0, 

I.e.,  when  x  =  2  or  when  a;  =  —  3 ; 

and  each  of   these  vahies  of   x,  when  substituted  in  the   given 
equation,  is  found  to  check. 

Ex.  2.    Solve  the  equation  x  (a;  —  3)  +  a;  +  2  =  2  (1  —  .-c^). 
Solution.     On  transposing,  etc.,  the  given  equation  becomes 
3ar'-2a;  =  0, 

i.e.,  a;(3a;-2)  =  0, 

wlience  x  =  0  or  -; 

3' 

and  these  values  of  x  are  found  to  check. 

EXERCISE  LXXXV 

Solve  the  following  equations  by  factoring,  and  check  the  roots 
in  each  case  : 

3.  /-5?/-24  =  0. 

4.  m--16  =  0. 

5.  a^  +  5x  =  21  +  x. 

6.  5  a;-=a^- 14. 

7.  5s-  =  Ss. 

8.  2v--30  =  9v-v- 

9.  2ar'-x=3. 

10.  4.  0^  =  9. 

11.  2c^  =  x(x-^r). 

12.  (4  x)-  =  14  (4  x)  — 

13.  22x  +  Sx'  =  4:x'- 

14.  ^-^"-  =  38. 


15. 

.t'--4a;  =  117. 

16. 

13y  +  2f=.5y-^4.y\ 

17. 

2iK2-20x  =  x- -51. 

18. 

3a(3a-l)=3a  +  24. 

19. 

2/-72/ 4-3  =  0. 

20. 

ic  (a;  +  7n)  =n{—x—m) 

21. 

~U0  +  x^=^-23x. 

22. 

7-2_5r  =  5r-25. 

23. 

Zaj^  —  Ikx  -}-  A:w  =  ma;. 

)•). 

24. 

5^2_3  =10^-3^1 

48. 

25. 

ao(?  -\-bx  —  ex. 

26. 

XT  _X        i»    ,     -j 

bo      be 

194  HIGH  SCHOOL  ALGEBRA  [Ch.  XII 

27.  If  a  quadratic  equation  in  one  unknown  number  has  no 
absolute  term,  show  that  one  root  of  the  equation  must  be  zero. 

125.  Completing  the  square.  What  must  be  added  to 
x^+Qx  to  make  it  the  square  oi  x+  S?  What  must  be  added 
to  m^—14:m  to  make  it  the  square  of  m  —  7  ? 

Since  Qx  ±k')^  =  x^  ±2kx  -\-  k^,  therefore  the  expression 
x^±2  kx,  whatever  the  value  of  k,  lacks  only  the  term  k^  of 
being  the  square  of  x±k;  hence,  if  the  square  of  half  the 
coefficient  of  the  first  power  of  x  he  added  to  an  expression  of 
the  form  x^  +  bx,  the  result  will  be  an  exact  square. 

Such  an  addition  is  usually  spoken  of  as  completing  the 
square. 

j^.^.,  if  (f  )2  is  added  to  «/2  +  5  ?/  it  becomes  (?/  + 1)2. 

126.  Solution    of    quadratics    by   completing  the  square.* 

There  are  many  quadratic  equations  which  cannot  easily  be 
solved  by  the  method  of  factoring  given  in  §  124.  All  quad- 
ratic equations,  however,  may  be  solved  by  the  method  of 
completing  the  square,  which  is  illustrated  below. 

Ex.  1.    Solve  the  equation  2a;^  —  3— 5a;  =  7a;+ll. 

Solution.     On  transposing,  etc.,  this  equation  becomes 

Now,  adding  9  to  each  member  (§  125,  and  Ax.  1),  we  obtain 
a^-6x  +  9  =  16, 
i.e.,  (a; -3)2  =  16, 

whence  (§  123)  a;  -  3  =  ±  4, 

i.e.,  a;  —  3  =  4  or  a;  —  3  =  —  4, 

and  therefore  a;  =  7  or  —  1. 

Moreover,  these  values  check,  and  are,  therefore,  the  required 
roots. 

*For  the  solution  of  quadratic  equations  by  means  of  a  formula,  see  §  178. 


124-126]  QUADRATIC  EQUATIONS  195 

Ex.  2.    Solve  the  equation  x^  +  11  a;  +  1  =  8  a:. 
Solution.     On  transposing,  the  given  equation  becomes 

a^-|-3ic  =  — 1, 
whence,  adding  (f )2,     x'-\-Sx  +  (^f  =  - 1  +  (f )',  [§  125 

i.e.,  (^-\-iy  =  h 

and  hence  .  a;  +  f  =  ±  V|=  ±  iV5,         [§121 

3^1    /-      -3±V5 

^  =  -2^2^^  =  ^ 

Moreover,  these  values  otx,  viz.,  ~'^^  ^  and  ~^~  ^,  check  (cf.  Ex. 
25,  p.  189),  and  are,  therefore,  the  required  roots. 

EXERCISE  LXXXVI 

3.    Solve  the  equation  ax^ -\-bx+  c  =0. 

Solution.    On  transposing  and  dividing  by  a,  this  equation  becomes 


whence 


a  a 

5  \2       12      c     b^-4ac 


a        \2al       ^a^     a  4  a^     '  '■* 

\        2a/  4  a^ 

therefore  ^  +  A  =  ±  J^!^  ^  ±  ^Z"' -  4  ac^     [5121 

2  a         ^      4  a^  2  a 

ig  ^^       b       ±Vb'^-4ac^-b±y/b'^-4:ac_ 

2a  2a  2a 

Moreover,  thesevaluesof  a;,  viz.,  -h  +  ^b^-4.ac  ^^^  -b-Vb^-4ac^ 

2a  2a 

check  (cf.  Ex.  29,  p.  189)  and  are,  therefore,  roots  of  the  given  equation. 

4.  What  must  be  added  to  each  of  the  following  expressions 
in  order  to  complete  the  square  (cf.  §  125):  x^-\-Sx',  P^  — 5P; 
(x  +  yy-4.(x  +  y)? 

5.  How  do  we  find  the  number  which  added  to  r^  +  ar  com- 
pletes the  square  ?     Explain. 


6. 

m^ -6  m  =40. 

7. 

y'-10y  =  75. 

8. 

W  =  2x-hx'. 

9. 

-S  =  2x^-h'^0x. 

10. 

a^  =  x-\-l. 

11. 

Sx'-2x  =  l. 

196  HIGH  SCHOOL   ALGEBRA  [Ch.  XII 

Solve  the  following  equations  by  the  method  of  completing  the 
square,  and  check  the  roots  in  each  case : 

20.  a!2-3a;-2  =  0. 

21.  ar^- 3  a;  H- 4  =  0. 

22.  |c^— 7c  =  c(cH-l). 

23.  6  +  ot  =  et\ 

24.  12a^-  x  =  6, 

25.  «-  =  -j6'  +  2. 

12.  2a;2-f3  =  7a;.  26.  r'-er=f. 

13.  3a^-10  =  7a;.  27.  3  x^ -\-5x-7  =x^-2  x. 

14.  (2  2/-3)2  =  6  2/  +  l.  28.  8m-10  =  3m^. 

15.  m(m  +  4)  =  7.  29.  A-a;-|ar^+2=0. 

16.  a2-6a  +  10  =  0.  30.  (y -Sf -4.(y-3)  =  117. 

17.  3  ('y^-'?;)  =2^2.^ 5  V +  4.       31.  (2m-3)2-6(mH-l)4-8=0. 

18.  y^  —  2cy  =  l.  32.  cV  +  2(Za;=— e. 

19.  r2  +  2ar  =  d  33.  (n +  1)2- 8  (n  +  1)  =  16. 

34.  Write  a  carefully  worded  rule  for  solving  such  quadratic 
equations  as  those  in  Exs.  6-33  above. 

35.  Show  that  the  rule  asked  for  in  Ex.  34  will  serve  to  solve 
such  an  equation  asa^  +  6i«  =  0.  Is  this  equation  more  easily 
solved  by  completing  the  square  or  by  factoring  ? 

127.   Avoiding  fractions   in    completing  the   square.     The 

method  employed  in  §  126  for  completing  the  square  often 
introduces  fractions  into  the  work,  and  these  sometimes  be- 
come troublesome  (cf.  Exs.  2  and  3,  p.  195).  A  method 
which  avoids  fractions  is  illustrated  below. 

Ex.  1.     Solve  the  equation  5  x^  —6x=  — 1. 

Solution.     On  multiplying  through  by  5,  we  obtain 
25aj2-30a;=-5, 
i.e.,  (5aj)2-6(5a;)  =  -5, 

whence,  adding  9,      (5  a;)^  —  6  (5  a;)  +  9  =  4, 
i.e.,  (5  X- 3)2  =  4; 


12(i-127J  QUADRATIC   EQlATIOlSfS  197 

therefore  5  ic  —  3  =  ±  2, 

from  which  5a;  =  3±2=:5orl, 

i.e.f  x  =  \  or  I ; 

and  each  of  these  values  is,  on  substitution,  found  to  check. 

Ex.  2.     Solve  the  equation  as?  -{-hxz^  —  c. 

Solution.     On  multiplying  through  by  4  a,  we  obtain 
4  a^ic-  +  4  abx  =  —  4  «c, 
i.e.,  (2  axf  +  2  6  (2  ax)  =  -  4  ae, 

whence,  adding  6^,      (2  aa;)^  +  2  6  (2  aa;)  +  6^  =  6^  —  4  ac, 
t.e.,  (2  ax  -j-  6)-  =  6^  _  4  ^f^ . 


therefore  2ax-\-h  =  ±  V  6^  —  4  ac, 


from  which  x  =  -l>±^h^-^  ^^^^  Ex.  3,  p.  195. 

•  2  a 

Note.  From  the  above  solutions  we  see  that,  if  an  equation  of  the  form 
aofi  +  &a;  +  c  =  0  is  multiplied  through  by  a  or  4  a,  according  as  b  is  even  or 
odd,  fractions  can  be  avoided  in  the  solution. 


EXERCISE  LXXXVII 

3.  What  must  be  added  to  each  of  the  following  expres- 
sions in  order  to  complete  the  square  :  4  x-  -f  8  x;  9  m*  +  12  m^; 
25  cH""  -  10  cd;  and  4  cV  -  4  cm  ? 

In  each  of  Exs.  4-11  belo^,  (1)  name  the  factor 'by  which  both 
members  of  the  equation  must  be  multiplied  if  fractions  are  to  be 
avoided  in  completing  the  square;  (2)  solve  the  equation  by  the 
method  of  §  127. 

4.  5 0^4-6 x  =  8.  8.  322  =  2  +  52. 

5.  3/  +  4?/  =  95.  9.  7  =  2x  +  3.^;^'. 

6.  2m2-f3m  =  27.  10.  6x^-x-^  =  0. 

7.  2^2^7^  +  6  =  0.  11.  15a;2-7a!-2  =  0. 

12.    By  the  method  of  §  127,  solve  Exs.  12-14  and  20-25,  p.  196. 


198  HIGH  SCHOOL   ALGEBRA    •  [Ch.  XII 

Solve  the  following  equations  by  the  method  of  §  127 : 

f^3t-{-5^     t  +  1  u'-^u     '^^  +  ^^0 

2  3  *        3     "^     4 

14.  ma^  — 6  a; +  3  =  0.  18.    3/  — 4%  +  2  =  0. 

15.  x^-\-px  +  q=:0.  19.    mx^ -^ nx -\- p  —  0. 

16.  mx^  =  2nx  —  k,  20.   ax^  —  5ax  =  a  — 11. 

128.  Fractional  equations  which  lead  to  quadratics.     As  in 

§  97,  so  here,  we  first  clear  the  given  equation  of  fractions, 
then  solve  the  resulting  integral  equation,  and  finally  check 
the  results  so  as  to  guard  against  the  introduction  of  extra- 
neous roots  (§  97,  note). 

Ex.  1.   Solve  the  equation  —^ — |- 1  =  3  a;. 

^  x  +  2 

Solution.    On  clearing  of  fractions,  etc.,  this  equation  becomes 
3  a;2  +  4  .T  -  7  =  0, 
whence,  solving  as  in  §  127,  we  obtain 

aj  =  l  or  —  1^; 
and  each  of  these  values,  when  substituted  in  the  given  equation, 
is  found  to  check. 

Ex.  2.   Solve  the  equation  -^—  _^4a;  +  3_    2  x" 


1  —  x       x-\-l        x^  —  1 
Solution.     On  clearing  of  fractions,  etc.,  we  obtain 
a^_2x-3  =  0, 
whence  a;  =  3  or  -  1,  [§  126 

of  which  3  checks,  but  —  1  is  extraneous  (§  97,  note). 

EXERCISE  LXXXVIII 

Solve  the  following  fractional  equations,  being  careful  to  ex- 
clude all  extraneous  roots : 

3.  15  a; +  -  =  11.  5. -  = -• 

X  bx-{-b      x-\-± 

4.  l-2  +  ..  =  2.  6.  -1-  + _!-  =  §. 
X                   X  1— sl+s3 


127-129]  qUADRATlG  EQUATIONS  199 

7  __3 L_  =  l.         11    2y  +  l      5^y-8 

'  2(a^-l)      4(aj  +  l)      8  ■  1-22/     7         2 

8  ^"^  I  ^  +  ^  =  2/^^"^^Y         12    ^^  +  ^  I  ct-2a;^22. 


07  +  2      a;— 2        V^~^/  2a  — x     a-{-2x 

9.  _l_  +  ^_=  J_.  13.   -*^  +  6  =  «J£+26). 
a;  —  1      a;— 2      3  —  a;  a  —  a;  a-{-b 

10.  ■20.+       40       ^_3M:7.  ^^_      .    __   ._.__,. 


:+3      s'+is-\-3            S  +  1                  X— 1       2  +  1 
15.^^  + ^ -=8+     ^ 


x  +  5      (a;  +  5)(a;  — 2)  a;  — 2 

129.   Problems  which  lead  to  quadratics.     As  in  §  50,  so 

here,  the  important  steps  in  the  solution  of  a  problem  are : 

1.  To  translate  the  verbal  language  of  the  problem  into 
algebraic  language,  i.e.,  into  equations. 

2.  To  solve  these  equations. 

3.  To  check,  and  interpret,  the  results. 

Special  emphasis  should  be  laid  upon  testing  and  interpret- 
ing results:  a  problem  often  contains  restrictions  upon  its 
numbers,  expressed  or  implied,  which  are  not  translated  into 
the  equations,  hence  the  solutions  of  the  equations  may  or 
may  not  be  solutions  of  the  problem  itself  (cf.  §  98). 

Prob.  1.  A  farmer  purchased  some  sheep  for  $  168,  and  later 
sold  all  but  four  of  them  for  the  same  sum.  If  his  profit  on 
each  sheep  sold  was  $  1,  how  many  sheep  did  he  buy  ? 

SOLUTION 

Let  X  =  the  number  of  sheep  purchased. 

Then  — -  =  the  number  of  dollars  each  sheep  cost, 

X 

1f>8 
and  =  the  number  of  dollars  received  for  each  sheep. 


A  I.                              168       168     ^ 
and  hence =  1 , 

X  —  4:  X 

therefore  (§  128)  a;  =  28  or  -24. 


"Profit  on  each  sheep 
being  $1 


200  BIGH  SCHOOL  ALGEBRA  [Ch.  XII 

The  first  of  these  values,  viz.,  28,  is  found  to  be  a  solution  of 
the  problem  as  well  as  of  the  equation,  but  while  the  second  satis- 
fies the  equation  it  cannot  satisfy  the  problem,  since  the  number 
of  sheep  purchased  is  necessarily  a  positive  integer. 

Prob.  2.  At  a  certain  dinner  party  it  is  found  that  6  times  the 
number  of  guests  exceeds  the  square  of  |  their  number  by  8; 
how  many  guests  are  there  ? 

SOLUTION  .       ^ 

Let  X  =  the  number  of  guests. 

Then  the  expressed  condition  of  the  problem  is 

«-(¥)'=»■ 

i.e.,  2x^-27  x  +  36  =  0, 

whence  x  =  12  or  |. 

Here,  too,  an  implied  condition  of  the  problem  is  that  the 
answer  must  be  a  positive  integer ;  hence  f ,  although  it  satisfies 
the  equation,  it  is  not  a  solution  of  the  problem. 

Prob.  3.  The  sum  of  the  ages  of  a  father  and  his  son  is  100 
years,  and  one  tenth  of  the  product  of  the  numbers  of  years  in 
their  ages,  minus  180,  equals  the  number  of  years  in  the  father's 
age ;  what  is  the  age  of  each  ? 

SOLUTION 

Let  X  =  the  number  of  years  in  the  father's  age. 

Then      100  —  x  =  the  number  of  years  in  the  son's  age, 
and  the  condition  of  the  problem  states  that 

whence  cc  =  60  or  30. 

Although  both  60  and  30  are  positive  integers,  yet  30  is  not  a 
solution  of  the  problem  :  it  would  make  the  son  older  than  the 
father.     Hence  the  father  is  60  years  old,  and  the  son  40. 

If,  in  the  above  problem,  "  two  persons  "  be  substituted  for  "  a 
father  and  his  son,"  etc.,  then  both  solutions  are  admissible,  and 
the  ages  are  either  60  and  40  years,  or  30  and  70  years. 


1210  QUADliATlC  EQUATIONS  201 

EXERCISE   LXXXIX 

4.  Divide  10  into  two  parts  whose  product  is  22f . 

5.  Find  two  numbers  whose  difference  is  11,  and  whose  sum 
multiplied  by  the  greater  is  513. 

6.  A  man  bought  a  flock  of  sheep  for  $  75.  If  he  had  paid 
the  same  sum  for  a  flock  containing  3  sheep  more  they  would 
have  cost  him  $  1.25  less  per  head.     How  many  did  he  buy  ? 

Is  each  solution  of  the  equation  of  this  problem  a  solution  of 
the  problem  itself?     Explain. 

7.  A  clothier  having  bought  some  cloth  for  $30  found  that  if 
he  had  received  3  yards  more  for  the  same  money,  the  cloth  would 
have  cost  him  50  cents  less  per  yard.  How  many  yards  did  he 
buy  ?     Has  this  problem  more  than  one  solution  ? 

8.  Find  two  numbers  whose  sum  is  10  and  whose  product 
is  42.     Can  these  numbers  be  real  (see  Note,  §  114)  ? 

9.  Find  two  consecutive  integers  the  sum  of  whose  squares  is 
61.  How  many  solutions  has  the  equation  of  this  problem  ? 
Show  that  each  of  these  is  a  solution  of  the  problem  also. 

10.  Are  there  two  consecutive  integers  the  sum  of  whose  squares 
is  118  ?  Are  there  two  numbers  whose  difference  is  1,  and  the 
sum  of  whose  squares  is  118  ?  What  are  they?  How  does  the 
second  of  these  questions  differ  from  the  first  ? 

11.  Find  three  consecutive  integers  whose  sum  is  equal  to  the 
product  of  the  first  two. 

12.  Is  it  possible  to  find  three  consecutive  integers  whose  sum 
equals  the  product  of  the  first  and  last  ?  How  is  the  impossibility 
of  such  a  set  of  numbers  shown  ? 

13.  In  selling  a  yard  of  silk  at  75  cents,  a  merchant  gains  as 
many  per  cent  as  there  are  cents  in  its  cost.     Find  the  cost. 

14.  A  cow  staked  out  to  graze  can  graze  over  a  circle  616  square 
feet  in  area ;  how  long  is  the  rope  by  which  she  is  tied  ?  [The 
area  of  a  circle  of  radius  r  is  tt  •  ?'^ ;  7r  =  3|,  approximately.] 


202  HIGH  SCHOOL   ALGEBRA  [Ch.  XII 

15.  Two  circles  are  such  that  the  difference  of  their  radii  is 
3  inches,  and  the  sum  of  their  areas  279|-  sq.  in.  Find  the  radius 
of  each  circle. 

16.  Find  two  numbers  whose  sum  is  |,  and  whose  difference  is 
equal  to  their  product.     How  many  solutions  has  this  problem  ? 

17.  The  product  of  three  consecutive  integers  is  divided  by 
each  of  them  in  turn,  and  the  sum  of  the  three  quotients  is  74. 
What  are  these  integers  ?  How  many  solutions  has  this  prob- 
lem ?     Explain. 

18.  If  the  product  of  two  numbers  is  6,  and  the  sum  of  their 
reciprocals  is  ff,  what  are  the  numbers  ?  How  many  solutions 
has  the  equation  of  this  problem  ?  How  many  solutions  has  the 
problem  itself  ?     Explain. 


19.  A  merchant  who  had  purchased  a  quantity  of  flour  for 
found  that  if  he  had  obtained  8  barrels  more  for  the  same  money, 
the  price  per  barrel  would  have  been  f  2  less.  How  many  barrels 
did  he  buy  ?     How  many  solutions  has  this  problem  ?     Explain. 

20.  Why  is  it  that  the  solutions  of  the  equation  of  a  problem 
are  not  always  solutions  of  the  problem  itself  ?     (Cf.  §  129.) 

21.  In  a  rectangle  whose  area  is  55i  sq.  in.,  the  sum  of  the 
length  and  breadth  is  15  in. ;  find  the  length. 

22.  Find  the  length  of  a  rectangle  whose  area  is  464  sq.  in., 
and  the  sum  of  whose  length  and  breadth  is  16  in. 

Interpret  the  imaginary  result  in  this  problem  (cf.  §  98).  Does 
an  imaginary  result  always  show  that  the  conditions  of  the  problem 
are  impossible  of  fulfillment  (cf.  Prob.  8,  p.  201)  ? 

23.  The  number  of  square  inches  in  the  surface  area  of  a  cube 
exceeds  the  number  of  cubic  inches  in  its  volume  by  8  times  the 
number  of  inches  in  one  edge.     Find  the  edge  of  the  cube.     How 

many  solutions  has  this  problem?     Explain. 

24.  In  the  trapezoid  ABCD,  whose  area  is 
75  sq.  in.,  the  altitude  BE  equals  BCy  and  AD 
is  5  in.  longer   than  BC  \   find  BC  and  AD. 


A    E 


^  [The  area  of  ABCD  =  \{BC-\-  AD)  •  BE.-] 


120 j  QUADRATIC    EQUATIONS  208 

25.  A  triangle  whose  base  is  2  in.  longer  than  its  altitude  has 
an  area  equal  to  that  of  a  rectangle  10  in.  by  4  in.  Find  the 
base  and  altitude  of  the  triangle.  [The  area  of  a  triangle  equals 
half  the  product  of  its  base  and  altitude.] 

26.  A  boating  club  on  returning  from  a  short  cruise  found  that 
its  expenses  had  been  $90,  and  that  the  number  of  dollars  each 
member  had  to  pay  was  less  by  4i  than  the  number  of  members  in 
the  club.     How  many  members  were  there  in  the  club  ? 

27.  If  in  Prob.  26  the  expense  of  the  cruise  had  been  $145 
the  other  conditions  remaining  unchanged,  how  many  members 
would  the  club  contain  ? 

What  is  the  significance  of  the  fractional  and  negative  results 
in  this  problem  ?  Do  such  results  always  indicate  that  the  con- 
ditions of  a  problem  are  impossible  of  fulfillment  ? 

28.  The  number  of  miles  in  the  distance  between  two  cities  is 
such  that  its  square  root,  plus  its  half,  equals  12.  What  is  this 
distance  ?     Has  this  problem  more  than  one  solution  ?     Explain. 

29.  When  a  certain  train  has  traveled  5  hours  it  is  still  60 
miles  from  its  destination.  If  by  traveling  5  miles  faster  per 
hour,  it  could  make  the  entire  trip  in  1  hour  less  than  the  sched- 
uled time,  find  the  entire  distance ;  also  the  actual  speed. 

30.  The  hypotenuse  of  a  right-angled  triangle  is  10  inches, 
and  one  of  the  sides  is  2  inches  longer  than  the  other ;  required 
the  length  of  the  sides  (cf.  Ex.  29,  p.  192). 

31.  It  took  a  number  of  men  as  many  days  to  dig  a  trench  as 
there  were  men.  If  there  had  been  6  more  men,  the  work  would 
have  been  done  in  8  days.     How  many  men  were  there  ? 

32.  A  crew  can  row  5|-  miles  downstream  and  back  again  in  2 
hours  and  23  minutes  ;  if  the  rate  of  the  current  is  3-^  miles  an 
hour,  find  the  rate  at  which  the  crew  can  row  in  still  water. 

33.  From  a  thread  whose  length  is  equal  to  the  perimeter  of  a 
square,  one  yard  is  cut  off,  and  the  remainder  is  equal  to  the 
perimeter  of  another  square  whose  area  is  -|  that  of  the  first. 
What  was  the  length  of  the  thread  at  first  ? 

HIGH    SCH.    ALG.  —  14 


204  HIGH  SCHOOL   ALGEBRA  [Ch.  XII 

34.  The  diagonal  and  the  longer  side  of  a  rectangle  are  to- 
gether equal  to  five  times  the  shorter  side,  and  the  longer  side 
exceeds  the  shorter  by  35  yards.  Find  the  area  of  the  rectangle. 
■  35.  A  ladder  13  ft.  long  leans  against  a  vertical  wall.  When 
the  distance  from  the  base  of  the  >vall  to  the  foot  of  the  ladder  is 
7  ft.  less  than  the  height  of  the  wall,  the  ladder  just  reaches  to 
the  top  of  the  wall.     How  high  is  the  wall  (cf.  Frob.  30)  ? 

36.  If  one  train,  by  going  15  miles  an  hour  faster  than  another, 
requires  12  minutes  less  than  the  other  to  run  36  miles,  what  is 
the  speed  of  each  train  ? 

37.  A  tank  can  be  filled  by  one  of  its  two  feed-pipes  in  2  hours 
less  time  than  by  the  other,  and  by  both  pipes  together  in  1|- 
hours.     In  what  time  can  each  pipe  separately  fill  the  tank  ? 

38.  The  owner  of  a  lot  56  rods  long  and  28  rods  wide  divided 
it  into  4  equal  rectangular  lots,  by  constructing  through  it  two 
streets  of  uniform  width.  If  these  streets  decrease  the  available 
area  of  the  lot  by  2  acres,  what  is  their  width  ? 

39.  One  of  two  casks  contains  twice  as  many  gallons  of  water 
as  the  other  does  of  wine ;  6  gallons  are  drawn  from  each  cask, 
exchanged,  and  emptied  into  the  other;  it  is  then  found  that  the 
percentage  of  wine  in  each  cask  is  the  same.  How  many  gallons 
of  water  did  the  first  cask  originally  contain  ? 

40.  A  and  B  together  can  do  a  given  piece  of  work  in  a  certain 
time  ;  but  if  they  each  do  one  half  of  this  work  separately,  A  works 
one  day  less,  and  B  two  days  more,  than  when  they  work 
together.     In  how  many  days  can  they  do  the  work  together  ? 

41.  In  going  a  mile,  the  hind  wheel  of  a  carriage  makes  145 
revolutions  less  than  the  front  wheel,  but  if  the  hind  wheel  were 

16  inches  greater  in  circumference,  it  would  then 
make  200  revolutions  less  than  the  front  wheel. 
What  is  the  circumference  of  the  front  wheel  ? 

42.  In  the  figure,  AB  =  BC=  CD  =  DA  =  10 
inches,  the  diagonals  AC  and  DB  bisect  each 
other  at  right  angles,  and  DB  is  4  inches  longer 
than  AC.  Find  the  lengths  of  AC  and  DB,  and 
the  area  of  the  figure. 


l-2i1-K30]  (QUADRATIC  KQrATIONS  205 

130.  Equations  in  quadratic  form.  Equations  which  con- 
tain onl}^  two  different  powers  of  the  unknown  number,  one 
of  these  powers  being  the  square  of  the  other,  are  said  to  be 
in  quadratic  form.  Thus,  a^+1  x^=S,  av?""  +  6m"  +  c  =  0, 
and  (2  s^  +  1)^  —  5  (2  s^  4. 1)  =  4  are  in  quadratic  form. 

Such  equations  may  be  solved  as  follows : 

Ex.  1.     Solve  the  equation  2y?{y? -\-X)=h —  oi?. 

Solution.     When  simplified,  the  given  equation  becomes 

or,  putting  y  for  a^,  2  2/^  +  3  ?/  —  0  =  0, 

whence  y=\  oy  -\,  [§126 

Li 

i.e.  (since  y  —  oc^),  a^  =  1  or  —  -f , 

whence  a;  =  ±  1  or  ±  V  — |. 

Moreover,  each  of  these  values  (1,   —1,  V— f,  and  —  V— f)* 
checks  (§  121),  and  is  therefore  a  root  of  the  given  equation. 


Ex.  2.     Solve  the  equation  Vx'  —  5  x  +  10  =  2  .'e^  — 10  aj  +  14. 
Solution.     This  equation  may  be  written  thus : 
Va;2-5a^  +  10  =  2(a.-2-5a;  +  10)-6; 
and,  on  putting  y  for  Vaf  —  5  ;r  +  10,  the  given  equation  becomes 

2/  =  2/-6, 
whence  y  =  2  ov—l,  [§  126 

i.e.,  Vaj^  —  5  a;  + 10  =  2  or  —  -|, 

and  therefore     aj'^  —  5  a;  + 10  =  4  or  f ,  [Squaring 

whence  a;  =  2,  3,  ^  ^  ~- ^  ^  or  ^-V^^  ^^  ^26 

all  of  which  values  check  (cf.  Ex.  28,  p.  189),  and  are  therefore  the 
required  roots. 

EXERCISE  XC 

Solve,  and  check  as  the  teacher  directs : 

3.  wt^  -16  =  0.  5.  ?/'  -  25  ?/2  +  144  =  0. 

4.  a;*_8x2  +  12  =  0.  6.  n*  =  18n2-32. 


206  BIGH  SCHOOL  ALGEBRA  CCh.  XII 


o       r»    /-  Hint.    Write  equation  thus : 

8.  x  =  3  —  2-Vx.  ^  , 

(2  k^  _  1)  -  6  \/2  A;-^  -  1  =  7. 

9.  20x'-23x'  =  -6.  , 

19.  y?-x-\-^x^—x-?,  =  ^. 

10.  4  —  ?ii^  =  18  m^. 

20.  5a;2  +  2V5a;2_a;  =  8  +  iK. 

11.  13  V^!  — 5  =  62;. 

.  1  21.    ?2^±2  +  --A_=2. 

13.   (x'  +  iy  +  4(a^  +  l)  =  45.         Hmx.    Let»:  =  <±2    j^^^i^y 


15.  x-2=Vi^+6.  23.    '^±5  +  ^^  =  7. 

r         r'  +  b 

16.  (m  +  l)-5Vm  +  l  =  6.  y  +  2      2(/  +  4)^51 

17.  2s-3  =  7V2i^^-12.  ^*'    /  +  4        2/  +  2         5* 


25.    5Vm2-10m  +  42  =  m2-10m  +  6. 


26.  ^2_7i_pVi2_7^^18^24. 

27.  a;4  +  4i»3-8x  +  3  =  0. 

Hint.  By  extracting  the  square  root  of  the  first  member  this  equation  may 
be  written  in  the  form  (x^  +  2  a;  -  2)^  =  1.    (Cf.  Ex.  32,  p.  183.) 

28.  2/'  +  22/^  +  52/'  +  42/  =  60. 

29.  16a;^-8aj3-31i»2  +  8a;  +  15  =  0. 

30.  9a;*  +  6a^-83a;2_28x  +  147  =  0. 

II.   SIMULTANEOUS  EQUATIONS   INVOLVING  QUADRATICS 

[Two  Unknown  Numbers] 

131.  One  equation  simple  and  the  other  quadratic.  Wlien 
one  equation  is  simple  and  the  other  quadratic,  we  may 
alwa.ys  eliminate  by  substitution  [§  103  (i)]. 

Ex.  1.    Solve  the  following  system  of  simultaneous  equations: 

(1) 

(2) 


I  3  a;  -  2  2/  =  3,    1 


l;jo-l3l] 


QUADRATIC  EQUATIONS 


3  +  2j 
»-       3       : 


Solution.     From  Eq.  (1), 
whence,  by  substituting  this  value  of  x,  Eq.  (2)  becomes 

and,  on  expanding  and  simplifying,  Eq.  (4)  becomes 

102/^  +  32/-27  =  0, 
whence  (§  126)  ^  =  f  or  - 1. 

Substituting  these  values  of  y  in  Eq.  (3)  shows  that 


207 
(3) 

(4) 


2/  =  |or  -f. 


according  as 

ia?  =  2,  ix  =  —  h 

and  \  g 

satisfy  the  given  system,  and  are  therefore  the  solutions  sought. 


EXERCISE  XCI 

Solve  the  following  systems  of  equations  and  check  your  results: 


6. 


'I 


8. 


x'-f  =  SO, 

9. 

x-\-y  =  10. 

x^-\.xy  =  12, 

x-y  =  2. 

10. 

3  uv  —  v—lOuj 

u-^2  =  V. 

4=x  +  Sy  =  9, 

11. 

2x'-{-5xy=:3. 

(a^  +  3)(2/-7)  =  48, 

12. 

x-{-y  =  lS. 

■st=A2, 

13. 

s-t  =  19. 

1  +  ^  =  10. 

14. 

f-f=^- 

2s  +  3^=10, 
8. 


f2s  +  3^  = 
\t(s  +  t)  = 


x^y     15' 
x—y  =  —  10. 
faj  +  f  2/  =  15, 
3ir-  — =  24. 

y 

1.5  ic—  .52/  =  6, 
.1  x^  -\-  .1  xy  =  16. 

16  +  4'y  +  2w2  =  5i^v, 
11  v  —  5  w  =  4. 

-H-l^  =  J  +  % 
xy  ar     ^/^ 

2-1  =  7. 
o;     y 


208  HIGH   SCHOOL  ALGEBRA  [Ch.  XII 

132.   Both    equations   quadratic :     one   homogeneous.     An 

equation  is  said  to  be  homogeneous  if  all  of  its  terms  are  of 
the  same  degree  in  the  unknown  numbers  (of.  §§  34,  93). 

If  one  of  two  given  quadratic  equations  is  homogeneous, 
the  system  may  always  be  solved  as  follows : 

Ex.  1.    Solve  the  following  system  of  equations  : 

6£c24-5a;2/-^6/  =  0,l  (1) 

2x2-2/2  +  5  0^  =  9.  J  (2) 

Solution.     On  dividing  Eq.  (1)  by  y^,  it  becomes 

whence  (§126)  ^^|or^^-|,  (4) 

i.e.,  x  =  lyov  x  =  -^y.  (5) 

On  substituting  the  first  of  these  two  values  of  x,  viz.,  ^y,  in 
Eq.  (2),  we  obtain 

2(t2/)^-/  +  5(f.v)=9,     • 
i.e.,  /- 30  2/4-81  =  0, 

whence  (§  124)  2/  =  ^^  or' 2/  =  3, 

and,  since  x  =  ^y,  the  coiTesponding  values  of  x  are  18  and  2. 

'x  =  2 


Moreover,  these  pairs  of  values,  viz.,  <j  '  and 

.y  =  *^' J 


J  are 

^  =  3, 


found  to  check,  and  are  therefore  solutions  of  the  system. 

Again,  by  substituting  in  Eq.  (2)  the  second  of  the  two  values 
of  X  in  Eq.  (5),  we  find  two  other  solutions  of  the  given  system 

VIZ.  \        ^  ^    and  \        ^' 

Note.  The  above  method  may  be  somewhat  simplified  by  substituting  a 
single  letter,  say  v,  for  the  fraction  x/y  in  Eq.  (3),  i.e.,  by  putting  x  —  vym 
the  homogeneous  equation.    Thus,  putting  vy  for  x,  Eq.  (1)  becomes 

6  i?2y2  4.  5  W2/2  -  6  2/2  =  0^ 
and  hence,  dividing  by  ?/2^  6  v^  +  5  u  —  6  =  0, 

whence  (§  126)  w  =  f  or  t7=  -f  ; 

and,  since  x  =  vy,  therefore  x  =  ^  y  or  x  =  —  ^  y.    From  here  on  the  work 
is  the  same  as  that  already  given, 


132-133] 


QUADRATIC  EQUATIONS 


209 


EXERCISE  XCII 

2.  Which  of  the  equations  in  Ex.  3-12  below  are  homogene- 
ous ?  Why  ?  Write  a  homogeneous  cubic  equation  involving 
the  unknown  numbers  r  and  s. 

By  the  method  of  Ex.  1  (or  of  the  note)  solve  the  following 
systems  of  equations ;  check  your  results  as  the  teacher  directs  : 


6. 


xy-\-f=2S, 

4:x'-27  xy  +  lSy^  =  0. 

x^  -\-xy  —  14:  =  y  —  Xj 

2  a^  —  3y^  =  xy. 

^(s  +  Q=36, 

b  s" -1^  St +  12  f  =  0. 


^ 


i3y?-5uv=:2v\ 
\u{u  —  v)  =  8. 


10. 


11. 


12. 


5a^-7a;2/-24  2/2  =  0, 

xy-ir2y'^z=5. 

aJ2/  +  3/-20  =  0, 

5x^  =  13xy  —^y^. 

8aj2  +  2/2  =  36, 
4a^-9a;2/  +  52/2  =  0. 
2{:»?  +  f)  =  nxy, 
x'-y'  =  75. 
5af  +  4:xy  =  y% 
x^-\-3x  =  5-{-y. 


Ex.  1.     Solve 


133.  Both  equations  homogeneous  except  for  the  absolute 
term.  A  system  consisting  of  two  quadratic  equations  each 
of  which  is  homogeneous  in  the  terms  containing  the  unknown 
numbers  can  be  solved  by  the  method  of  §  132,  Note. 

x^  +  3xy  +  2f  =  S,  (1) 

xy-4:f  =  2.  (2) 

Solution.  By  substituting  vy  for  x  in  Eqs.  (1)  and  (2)  we 
obtain  3  vY  +  3  vy^ -{- 2  y^  =  S,  (3) 

and  vY  —  vy^  —  4:y^  =  2,  (4) 

whence,  from  (3)  and  (4),  respectively, 

2 


\x' 


y 


therefore 
whence 


3  -y^  +  3  v  +  2 
8 


and  2/^ 


V' 


4' 


3v^  +  3v  +  2       'y2_-y_4' 

v  =  -  2  or  9. 


(5) 
(6) 


210 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XII 


Hence,  from  (5), 


I.e. 


if  =  l  or/=3\,  ^ 
2/  =  1  or  —  1  or  2/  =  +  V  3^; 


34 


or 


V,>j, 


and  substituting  these  values  of  y  and  v  in 

x=vy, 
we  obtain,  as  corresponding  values, 

a?  =  —  2  or  4-2,  and  also  +  9  V^^^  or  —  9  V^; 
and,  checking,  we  find  the  solutions  of  the  given  system  to  be  : 

Note.  The  success  of  the  '■'•vy  method"  employed  above  is  due  to  the 
fact  that  by  eliminating  the  absolute  term  from  the  given  system  of  equations 
we  obtain  a  homogeneous  equation  (cf.  §  132). 

EXERCISE  XCIII 

Solve  the  following  systems,  and  check  your  results: 


2. 


3. 


6. 


7. 


ar^  +  2/2  =  29, 
xy  =  10. 
m^  —  mn  =  8, 
mn  -\-n^  =  12. 
af  —  Sxy  =  — 14, 
xy-\-2f  =  S9. 
f  +  2xy  =  -% 
x^  —  xy=z70. 
2x^-xy  =  2S, 
a^  +  22/'  =  18. 
?/2  4. 15  =  2  xy, 
x'  +  y'  =  21  +  xy. 


8. 


10. 


11. 


12. 


x^  +  xy  —  40, 

27  +  2y'-3xy  =  0. 

2a?-  +  3a!?/  +  /  =  20, 
5a;2  +  42/'  =  41. 

xF  —  xy  —  y^=  20, 
a;2_3(»2/  +  22/2  =  8. 
i*^  H-  3  wv  +  'y^  =  61, 
u^-v^^Sl-2uv. 
3f_5y2x_        ^ 


4  iC 

~3" 


'^  _y^  +  2xy 
l~2{l-x)' 


13.    Solve  Ex.  1  by  eliminating  the  absolute  term  and  then 
applying  §  132. 

By  the  method  suggested  in  Ex.  13,  solve : 

x'-3xy  +  4:f=:Sy,  ^^     i3x'-5  xy -4.f  =  Sx, 

x^-4:xy-^2y^  =  2y.  '    [9  a^ +  xy- 2  y^=6  x. 

x'-4.xy-^3f  =  -3y,         ^^     Ux' -^6xy -f  =  ^,y, 
3x^  —  5xy  =  6y.  \6x-—9xy  +  2y-  =  2y. 


14. 


15. 


138-134]  QUADRATIC  EQUATIONS  211 

134.  Special  devices.  The  kinds  of  systems  of  equations 
specified  in  §§  131-133  occur  frequently,  and,  although  they 
present  themselves  in  a  great  variety  of  forms,  they  may 
always  be  solved  by  the  methods  there  given. 

Special  devices  for  elimination,  however,  often  give  sim- 
pler and  more  elegant  solutions;  some  of  these  devices  are 
illustrated  below. 

(i)  Solving  hy  first  finding  x-\-  y  and  x  —  y, 

x-y  =  5,  (1) 


Ex.  1.    Solve  the  equations  , 

[xy  =  -6.  (2) 

Solution.     From  (1),  x"  -  2xy -^y^  =  25,  (3) 

and  from  (2),                                        4  a;?/  =  -  24 ;  (4) 

adding  (4)  to  (3),                x^-\-2xy-{-  y'  =  1,  .  (5) 

whence                                               x-\-y  =  ±lj  (6) 

and  from  (1)  and  (6),  «  =  3  or  2. 

The  corresponding  values  of  y  are        y  =  —2ov  —  3 ; 

.     •  P  1       .  (x  =  3,  (x  =  2, 

i.e.,  the  solutions  of  the  ffiven  system  are:  J  ^   and  J 

^  ^  \y  =  -2,  l.y  =  -3. 

(a;2  +  2/'  =  5,  (1) 

Ex.  2.     Solve  the  equations  i    „  ,      „ 

^  [x'-xy  +  f^^-  (2) 

Solution.     Subtracting  (2)  from  (1),  xy  =  2,  (3) 

whence,  adding  2  •  (3)  to  (1),  x''  +  2  xy  +  2/^  =  9,  (4) 

and  subtracting  2  •  (3)  from  (1),      x^ —  2xy -{-y'^  =  l\,  (5) 

therefore,  from  (4),  a;  +  ?/  =  ±  3,  (6) 

and  from  (5),  x  —  y  =  ±l]  (7) 

from  (6)  and  (7),  we  now  easily  find  the  following  solutions : 

\y  =  l,         \y  =  2,         U  =  -l,  U  =  -2, 

all  of  which  check. 


212  HIGH  SCHOOL   ALGEBRA  [Ch.  XII 

(ii)  Solving  hy  dividing  one  equation  hy  the  other. 

a^^-/  =  3,  (1) 


Ex.  3.     Solve  the  equations   . 

'^-2/=l.  (2) 

Solution.     On  dividing  (1)  by  (2),  member  by  member,  we 
obtain  x-\-y  =  d,  (3) 

whence,  from  (2)  and  (3),  a;  =  2,  and  y  =  l. 

ar^  +  ^«  =  26,  •    (1) 


Ex.  4.     Solve  the  equations   , 

^  [x  +y=   2.  (2) 

Solution.  On  dividing  (1)  by  (2),  member  by  member,  we 
obtain  x^  —  xy-^y^  =  13,  (3) 

and  (2)  and  (3)  may  now  be  solved  either  like  Exs.  1  and  2  above, 
or  by  the  method  of  §  131. 

(iii)  Solving  by  considerations  of  symmetry.  An  equation 
is  symmetric  with  regard  to  two  of  its  letters  if  it  is  not 
changed  by  interchanging  those  letters.  Thus  :  x  +  y  =  S^ 
and  §2  ^  st  -\- 1^  =  5  (^s  -{- f)  are  symmetric  equations. 

Two  equations  which  are  symmetric  (or  symmetric  except 
for  the  signs  of  one  or  more  terms)  may  often  be  solved  by 
substituting  u  +  v  and  u  —  v,  respectively,  for  their  unknown 
numbers. 

^  +  2/^=6,  (1) 


Ex.  5.     Solve  the  equations 

^xy  =  2(x^y)-5.  (2) 

Solution.     On  putting  x  =  it-{-v  and  y  =  u  —  v,  the  given  equa- 
tions become,  respectively, 

2u^  +  2v^  =  6,  Sindu^-v^  =  4:U-5',  (3) 

therefore,  eliminating  v^  and  simplifying, 
u'-2u  +  l  =  0, 
whence  u  =  l. 

Substituting  this  value  of  u  in  either  one  of  Eqs.  (3),  gives 

v=±V2, 
whence  (since  x  =  u-^  v,  and  y  =  u  —  v), 

x=l±  V2,  and  y~lT  V2. 


134]  QUADRATIC  EQUAIIONS  213 

( oc^  -^  Tf  =  xy  —  5f  (1) 

Ex.  6.     Solve  the  equations  {  ^      ^ 

\x-^y  +  l  =  0.  (2) 

Solution.     On  putting  x  =  u-\-v  and  y  =u  —  v,  (1)  and  (2)  be- 
come, respectively, 

2u^  +  Quv'-u^  +  v^  +  b  =  0,  (3) 

and  2  2^  + 1  =  0.  (4) 

From  (4)  u  =  —  |, 

and  substituting  this  value  in  (3)  gives 

whence  a;  =  1  or  —  2,  and  2/  =  —  2  or  1. 

EXERCISE  XCIV 

By  the  method  of  134  (i)  solve  the  following  systems : 
ja^  +  /  =  13,  |a^  +  2/'  =  l, 

^'    \xy  =  ^.  ^^'    |25a^y  +  12  =  0. 


1^2  +  ^2^61, 
'm4-n  =  24. 


[5mn-2  =  0. 


mn         Q  |a;2  +  2/2  =  a, 

— y.  12. 


9  [a;-f-y  =  6. 

By  the  method  of  134  (ii)  solve  the  following  systems : 

13-  „  15.  ^ 

\r  —  s  =  l.  [r— p  =  7. 


14.    \  16 

r4-s=  7. 


aj^  +  2/3  =  a, 
a:  +  2/  =  6. 
By  the  method  of  134  (iii)  solve  the  following  systems : 

17.  |-^  +  -"-*  =  26.  f2(.  +  ,)  =  -§f^, 
I  a; +  2/ =  6.                             20.    ]  5 

a: -2/ =  7. 

18.  \     ■    -2/)  =  -^^2/'  f2x2_      _^22/^  =  62, 
2.                            21.    {  _  ^. 

[  a;  —  2/ —  o  0^2/ =  ~~  "1- 

^^-    [ar^  +  2r''  =  37.  ^^'    1^  +  2.9/4-^-22. 


^     |3(a;-2/ 
[a;  +  2/  = 


214 


HIGH   SCHOOL  ALGEBBA 


[Ch.  XII 


Solve  the  following  systems  of  equations,  choosing  for  each 
the  method  (§§  131-134)  which  seems  to  you  best: 


23. 


24. 


25. 


26. 


27. 


28. 


29. 


30. 


31. 


^_?/_16 
y     X     lo' 
a;-2/  =  2. 

xF  —  xy  =  6, 
a;2-f2/2  =  61. 

^+^.  =  74, 
a;-      2/    . 

1-1  =  2. 


rcf.    Ex."! 
13,  §104.  J 


;i-^      y^ 

i+l= 


=  91, 

7. 


•1  +  1-1 

X      y      Z 

.^2/      18 

a;  +  ?/  .X  —y  _  10 
a;  —  ?/      x-{-y      3' 

a;^  +  2/^  =  45. 

1       1 


=  119, 


-7  =  i. 


aJ^  +  2/^  +  i»  =  2/  +  14:, 
a;?/  =  6. 

=  96  —  4mn , 
6. 


m  -\-n 


33. 


34. 


35. 


36. 


37. 


38. 


x-^y  =  25, 

V^+  ■Vy  =  7. 


a^  +  2/  +  2Va;  +  2/  =  24, 
a;  —  ^  -f-  3Va/'  — 2/  =  10. 

2(a^2  +  /)=o.r2/, 


x^-2xy  =  S  y% 
y{x  +  y)=4.. 

(2+^(2/  +  l)=4, 
V2  +  aj-V//  +  l  =  -J-. 

{a'  +  h'  =  n, 

|a262  =  4. 


39.    I 


40. 


41. 


42. 


43. 


44. 


45. 


a^  4-  2/^  +  6  Va;^  +  2/^  =  So^ 
a;2_2/2  =  7. 

tv=-  26. 

^'  +  sV  +  ^^  =  9, 

s2  +  6-^  +  2/'  =  3. 

s^-t^  =  37, 
st(s-t)  =12. 
x^  +  2/^  =  97, 
a;  +  2/  =  — 1- 

o  ,    3  771        f. 

3  mn  H =  o, 

n 

3^n  +  — =  2.5. 
m 

2/       a;      ' 


i;U]  qUADHATIC  EQUATIONS  215 

PROBLEMS 

1.  The  sum  of  two  numbers  is  14,  and  the  difference  of  their 
squares  is  28.     What  are  the  numbers  ? 

2.  Find  two  numbers  whose  difference  is  15,  and  such  that  if 
the  greater  is  diminished  by  12,  and  the  smaller  increased  by  12, 
the  sum  of  the  squares  of  the  results  will  be  261, 

3.  Find  two  numbers  whose  difference  is  80,  and  the  sum  of 
whose  square  roots  is  10. 

4.  Given  that  one  root  of  a  quadratic  equation  is  4  times  the 
other  and  that  their  product  is  ^,  find  the  roots ;  then  form  the 
equation  which  has  these  roots  (cf.  §  72). 

5.  The  sum  of  the  roots  of  a  quadratic  equation  is  12,  and 
their  product  is  —  189.     What  is  the  equation  ? 

6.  The  sura  of  two  numbers,  their  product,  and  also  the  dif- 
ference of  their  squares  are  all  equal ;  find  the  numbers. 

7.  If  the  length  of  the  diagonal  of  a  rectangular  field,  contain- 
ing 30  acres,  is  100  rods,  how  many  rods  of  fence  will  be  required 
to  inclose  the  field  ? 

8.  Find  the  dimensions  of  a  rectangular  field  whose  perimeter 
is  188  rods  and  whose  area  "will  remain  unchanged  if  the  length 
is  diminished  by  4  rods  and  the  width  increased  by  2  rods. 

9.  The  sum  of  the  circumferences  of  two  circular  flower  beds 
is  56|^  feet,  and  the  sum  of  their  areas  is  141^  square  feet.  Find 
the  radius  of  each.     (Cf.  Ex.  14,  p.  201.) 

10.  A  circular  table  whose  radius  is  3J-  feet  has  the  same  area 
as  a  rectangular  table  whose  length  is  5  inches  more  than  its 
breadth.     Find  the  dimensions  of  the  rectangular  table. 

11.  A  sum  of  money  lent  at  a  certain  rate  of  interest  gives  an 
annual  income  of  $  450 ;  if  the  sum  were  ^  500  more  and  the 
rate  1  %  less,  the  annual  income  would  be  $  50  less.  Find  the 
principal  and  the  rate. 

12.  A  sum  of  money  at  interest  for  one  year  at  a  certain  rate 
amounted  to  $11,130.  If  the  rate  had  been  1  %  less  and  the 
principal  $100  more,  the  amount  would  have  been  the  same. 
What  was  the  principal,  and  what  the  rate? 


216  IHGU  SCHOOL   ALdKliRA  [Ch.  XII 

13.  A  formal  rectangular  flower  garden  is  to  be  enlarged  by  a 
border  whose  uniform  width  is  10  %  of  the  length  of  the  garden. 
If  the  area  of  the  border  is  900  sq.  ft.,  and  the  width  of  the  old 
garden  is  75  %  of  the  width  of  the  new  one,  find  the  dimensions 
of  the  garden  and  the  width  of  the  border. 

14.  A  certain  kind  of  cloth  loses  2  %  in  width  and  5  %  in 
length  by  shrinking.  Find  the  dimensions  of  a  rectangular  piece 
of  this  cloth  whose  shrinkage  in  perimeter  is  38  in.,  and  in  area 
8.625  sq.  ft. 

15.  The  perimeter  of  a  right-angled  triangle  is  24  ft.,  and  its 
area  is  24  sq.  ft.     Find  the  length  of  each  side  in  the  triangle. 

16.  In  the  right-angled  triangle  ABC,  BD  is 
drawn  perpendicular  to  AC.  If  BC  =12,  AC  =11, 
and  BD=^AD'DC,  find  BD,  AD,  and  DC 

17.  The  combined  capacity  of  two  cubical  coal 
bins  is  2728  cu.  ft.,  and  the  sum  of  their  lengths 
is  22  ft.;  find  the  length  of  the  diagonal  of  the 
smaller  bin. 

18.  Find  two  numbers  whose  product  is  8  greater  than  twice 
their  sum,  and  48  less  than  the  sum  of  their  squares. 

19.  Find  two  numbers  such  that  the  sum  of  their  fourth  powers 
is  881  while  the  sum  of  their  squares  is  41. 

20.  The  total  area  of  the  walls  and  ceiling  in  a  room  9  ft. 
high  is  575  sq.  ft.  Find  the  length  and  breadth  of  the  room 
if  their  sum  is  24  ft. 

21.  A  farmer  found  that  he  could  buy  16  more  sheep  than 
cows  for  $  100,  and  that  the  cost  of  3  cows  was  $  15  greater  than 
the  cost  of  12  sheep.     What  was  the  price  of  each  ? 

22.  If  5  times  the  sum  of  the  digits  of  a  certain  two-digit 
number  is  subtracted  from  the  number,  its  digits  will  be  inter- 
changed; and  if  the  num.ber  is  multiplied  by  the  sum  of  its  digits, 
the  product  will  be  648.     What  is  the  number  ? 

23.  Find  two  numbers  such  that  the  square  of  either  of  them 
equals  112  diminished  by  12  times  the  other. 


1;)4-135J  QUADRATIC    EQUATIONS  217 

24.  If  5  is  added  to  the  numerator  and  subtracted  from  the 
denominator  of  a  given  fraction,  the  result  equals  the  reciprocal 
of  the  fraction ;  and  if  2  is  subtracted  from  the  numerator,  the 
result  equals  J  of  the  original  fraction.     Find  the  fraction. 

25.  The  distance  (s)  in  meters,  through  which  a  body  falling 
from  a  position  of  rest  passes  in  the  ^th  second  of  its  fall  is 
given  by  the  formula  s  =  \g  (2t—l)  ;  and  the  total  distance  (S) 
fallen  in  t  seconds  is  S=^gt^.  How  long  has  a  body  been  falling 
when  s  =  44.1  meters  and  S  =  122.5  meters?  If  g  is  less  than  10, 
what  is  its  value? 

26.  Solve  the  problem  of  Ex.  25  if  s  and  S  are  each  expressed 
in  feet,  and  s  =  112^7^  and  S  =  2571 

27.  In  going  40  yards  more  than  ^  of  a  mile  the  fore  wheel  of 
a  carriage  revolves  24  times  more  than  the  hind  wheel ;  but  if  the 
circumference  of  each  wheel  were  3  ft.  greater,  the  fore  wheel 
would  revolve  16  times  more  than  the  hind  wheel.  What  is  the 
circumference  of  the  hind  wheel  ? 

28.  A  merchant  paid  $125  for  an  invoice  of  two  grades  of 
sugar.  By  selling  the  first  grade  for  $91,  and  the  second  for 
$36,  he  gained  as  many  per  cent  on  the  first  grade  as  he  lost  on 
the  second.     How  much  did  he  pay  for  each  grade  ? 

29.  Two  trains  start  at  the  same  time  from  stations  A  and 
B,  320  miles  apart,  and  travel  toward  each  other.  If  it  re- 
quires 6  hr.  and  40  min.,  from  the  time  the  trains  meet,  for  the 
first  train  to  reach  B,  and  2  hr.  and  24  min.  for  the  second  to 
reach  A,  find  the  rate  at  which  each  train  runs. 

30.  After  traveling  2  hr.,  a  train  is  detained  1  hr.  by  an  acci- 
dent ;  it  then  proceeds  at  60  %  of  its  former  rate,  and  arrives 
7  hr.  40  min.  late.  Had  the  accident  occurred  50  miles  farther 
on,  the  train  would  have  been  6  hr.  20  min.  late.  Find  the 
distance  traveled  by  the  train.     (Cf.  Ex.  42,  p.  164.) 

135.*  Simultaneous  quadratics  not  always  solvable  by  methods 
already  given.     While  many  systems  containing  quadratics  (and 

*  §  135  may,  if  the  teacher  prefers,  be  omitted  till  the  subject  is  reviewed. 


'21S  HIGH   SCHOOL   ALGEBRA  [Ch.  XIl 

some  containing  still  higher  equations)  may  be  solved  by  the 
methods  of  §§  131-134,  these  methods  do  not  always  suffice  for 
the  solution  of  such  systems. 

Thus,  inspection  shows  that  the  system 
a^-3a;  +  82/  =  4, 
3a;^-16/  +  20  2/  =  9, 
cannot  be  solved  by  the  methods  of  §§  132-134;  and  elimination 
by  substitution  (as  in  §  131)  leads  to  an  equation  of  the  fourth 
degree  in  one  unknown  number,  viz.,  to 

a,>4  _  6  ar^-i»2- 6  a; +12  =  0, 
which  cannot  be  solved  by  the  elementary  methods  already  studied. 
Such  equations  are  discussed  in  higher  algebra. 

For  all  systems  like  the  above,  however,  approximate  solutions 
may  be  obtained  by  means  of  graphs  (cf.  §  143). 

136.*  Systems  containing  three  or  more  unknown  numbers.  Some 
systems  containing  three  or  more  simultaneous  equations,  some 
of  which  are  quadratic,  may  be  solved  by  elementary  methods. 

E.g.,  if  one  equation  of  a  given  system  is  quadratic,  and  all  the 
others  are  of  the  first  degree,  then  a  slight  modification  of  the 
method  of  §  131  will  provide  a  solution  (cf.  El.  Alg.  §  180). 

The  solution  of  such  systems  in  general  is,  however,  beyond 
the  limits  of  this  book. 

*  This  article  may,  if  the  teacher  prefers,  be  omitted  till  the  subject  is 
reviewed. 


CHAPTER   XIII 


GRAPHIC  REPRESENTATION  OF  EQUATIONS* 

137.  Introductory.  Although  an  equation  in  two  unknown 
numbers  has  (§  99)  an  infinitely  large  number  of  solutions, 
and  is  in  that  sense  indeterminate,  yet  by  a  beautiful  device, 
due  to  the  celebrated  mathematician  and  philosopher  Des- 
cartes (pronounced  da-kart',  born  1596,  died  1650),  a  perfectly 
definite  picture  of  such  an  equation  may  be  made  (cf.  §  139). 

138.  Axes.  Coordinates.  Let  us  draw  (as  Descartes  did) 
two  perpendicular  straight  lines  X'X 

and  Y'  Z",  cutting  each  other  in  the 
point  0,  and  call  these  lines  the 
coordinate  axes.  If  we  now  agree 
that  distances  measured  toward  the 
right  from  Y'Y^  or  upward  from 
X'X^  shall  be  positive,  while  dis- 
tances toward  the  left,  or  downward, 
shall  be  negative,  then  any  point  in 
the  plane  of  this  page  can  be  located 
as  soon  as  we  know  its  distances  from  the  axes  X' X  and  Y'  Y. 

Thus,  to  locate  the  point  P,  3  inches  from  F'^and  2 
inches  from  X' X^  we  measure  off  3  inches  (represented  in  the 
diagram  by  3  spaces)  toward  the  right  from  0,  to  the  point 
M^  say,  and  then  2  inches  upward  from  M. 

The  numbers  which  serve  to  locate  a  point  (in  this  case 


1 

( 

p 

— -1 

, 

r 

0 

J 

M 

- 

- 

- 

p? 

/ 

1 

*This  chapter  should  he  included  in  the  course  whenever  possible 
omission,  however,  will  not  break  the  continuity  of  the  work. 

HIGH    SCH.   ALG. — 15  219 


its 


220 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XIII 


3  and  2)  are  called  the  coordinates  of  the  point.     The  point 
P  may  be  represented  by  the  symbol  (3,  2). 

Similarly  the  point  ^  (—  3,  4)  is  located  by  measuring  3 
spaces  toward  the  left  from  0,  and  then  four  spaces  upward. 
The  point  i2  (—  2,  —  3),  also,  is  represented  in  the  figure. 

Note.  This  plan  of  locating  points  in  the  figure  somewhat  resembles  that 
used  to  locate  places  on  the  earth's  surface  by  their  latitude  and  longitude. 
The  coordinate  axes  correspond  to  the  equator  and  the  prime  meridian. 


y 

1 

p 

X 

, 

^ 

X 

0 

Q 

s 

, 

1 

EXERCISE  XCV 

1.   Name   the   ic-coordinate    (i.e.,   the   distance  from   the  axis 
Y'Y)  of  each  point  located  in  the  figure  below.     Also  name  the 
^/-coordinates  of  these  points. 

Draw  a  pair  of  axes  as  in  §  138  and 
locate  the  following  points : 

2.  (5,4);  (3,7);  (4,-2);   (-3,1); 
and  (-4,  -6). 

3.  (-i,l);  (i,|);  (H,  -3);  (4,1); 
and  (  — I,  —5). 

4.  (3,  0);    (-5,  0);    (0,  8);    (0,  0) ; 
and  (0,  -2). 

5.  Where  are  the  points  whose  ^/-coordinate  is  0  ?  Where  are 
those  whose  ^-coordinate  is  0  ?     those  whose  ^/-coordinate  is  3^  ? 

6.  Locate  five  points  each  of  which  has  its  a7-coordinate  equal 
to  its  ^/-coordinate,  and  draw  a  line  through  these  points.  Does  this 
line  contain  any  other  points  whose  two  coordinates  are  equal  ? 

7.  Where  are  the  points  which  have  their  ^/-coordinates  oppo- 
site in  value  to  their  respective  ic-coordinates  ? 

8.  Verify  that  the  equation  2  x  —  y==3  is  satisfied  by  each  of 
the  following  number-pairs  :  (0,  —  3)  ;  (1,  —  1)  ;  (2,  1)  ;  (3,  3)  ; 
(4,  5) ;  then  locate  the  point  corresponding  to  each  pair.  On 
what  kind  of  a  line  do  these  points  lie  ? 

9.  Measure  the  coordinates  of  several  other  points  on  the  line 
mentioned  in  Ex.  8.    Are  these  coordinates  solutions  of  2  x—y=3? 


i;W-140]        (lliAPlIJC  liEPIiESKNTATION  OF  EQUATIONS       221 

10.  Locate  the  following  points :  (0,  5) ;  (0,  —  5) ;  (5,  0) ; 
(-5,0);  (4,3);  (-4,3);  (4,-3);  (-4,-3);  (3,4);  (-3,4); 
(3,  —4);  (  —  3,  —4).     On  what  kind  of  a  line  do  they  lie? 

139.  The  picture  (graph)  of  an  equation.  Consider  the 
equation  2  a;  —  y  =  3. 

This   equation  is,  manifestly,  satisfied   by  the   following 
pairs  of  values  of  x  and  7/  (§  99)  : 
(-1,  -5);  (0,  -3);   (1,  -1);  (2,  1);   (3,  3);    (4,5);   etc. 

If  we  now  locate  (as  in  §  138)  the  points  A,  B,  (7,  etc., 
corresponding  to  these  number-pairs,  we  find  that  they  are 
not  scattered  at  random  over  the 
page,  but  that  the^  all  lie  upon  the 
straight  line  US  in  the  figure  (cf. 
Ex.  8,  p.  220).  Moreover,  the  co- 
ordinates of  every  point  in  the  line 
RS,  and  those  of  no  other  points 
whatever,  satisfy  the  given  equa- 
tion.* 

For  these  reasons,  the  line  MS  may 
be  regarded  as  the  picture  of  the 
equation  ;  it  is  usually  called  the 
graph,  also  the  locus  of  the  equation.  That  is,  the  graph 
(or  locus)  of  an  equation  is  the  line  (or  lines)  containing 
all  the  points  (and  no  others)  whose  coordinates  satisfy  the 
given  equation. 

140.  Drawing  of  graphs.  The  method  illustrated  in  §  139, 
for  finding  the  graph  of  an  equation  in  x  and  «/,  may  be 
stated  thus  : 

(1)  Solve  the  given  equation  for  ?/,  in  terms  of  x. 


V                 /a 

7 

t 

7 

T 

X          o~r^      zE. 

/ 

M 

t 

i 

f 

^ 

t. 

-hi     v^ 

^          r 

*  Let  the  pupil  test  this  statement  by  careful  measurement  on  a  large  and 
well-drawn  figure.  The  proof  of  its  correctness  follows  easily  from  the 
theory  of  similar  triangles  in  geometiy. 


•>•^:> 


Hid  1 1   S(1I(K)L   ALGEBRA 


[Ch.  XIII 


(2)  Assign  to  rr  a  succession  of  values,  such  as  0,  1,  2, 
3,  •••  (also  —1,  —2,  —  3,  •••),  and  find  the  corresponding 
values  of  y ;  ^.e.,  find  a  succession  of  solutions  of  the  given 
equation. 

(3)  By  means  of  a  pair  of  axes  locate  the  points  corre- 
sponding to  these  solutions,  —  use  cross-section  paper. 

(4)  Draw  a  line  connecting  these  points  in  regular  order  ; 
this  line  is  (approximately)  the  graph  of  the  given  equation. 

E.g.,  to  find  the  graph  of  the  equation  3y—x^  =  0,  we  solve 
the  equation  for  y  in  terms  of  x,  and  tabulate  the  corresponding 
values  of  x  and  y,  thus : 

Locating  the  points  0,  A,  B,  C, 
etc.,  and  connecting  them  in  order, 
we  obtain  the  line  NML  •••£/,  which 


X-. 


X 

y 

Points 

0 

0 

0 

1 

i 

A 

2 

1 

B 

3 

3 

C 

-1 

i 

H 

—  2 

i 

K 

-3 

3 

L 

• 

: 

' 

II  ITT 

~E                 ji  T 

j-i^                      4 

t       "             7 

\                     t 

\                   I 

-M^                              J? 

A       i^    t 

->            f- 

X            ~K- 

%        t 

\      / 

T '        k\      /(^         t. 

^             Tt-  -^               ^ 

±        :£  ^ 

vr' 

is  a  good  approximation  to  the  graph 
of  the  given  equation. 
By  assigning  to  x  values  between  0  and  1,  1  and  2,  etc.,  and 
finding   the    corresponding    values   of  y,    we   can   locate   points 
between  0  and  A,  A  and  B,  etc.,  and  thus  draw  a  closer  approxima- 
tion to  the  required  graph. 

Note.  The  graph  of  a  first  decree  equation  in  x  and  y  is  (cf.  §  l.SO)  a 
straight  line  (hence  a  first  degree  equation  is  often  called  a  linear  equation). 
In  this  case,  of  course,  only  two  points  (i.e.,  two  solutions  of  the  equa- 
tion) need  be  found  in  order  to  draw  the  complete  graph. 


UO-Ulj      GRAPHIC  REPRESENTATION  OF  EqUATlONti       228 


EXERCISE  XCVI 

1.  Find  six  solutions  of  2x-{-y  =  12,  locate  the  points 
determined  by  these  solutions,  and  draw  the  graph  of  the  equation. 

Using  the  plan  of  Ex.  1,  draw  the  graph  of : 

2.  a;  +  2^  =  8.  5.    2a;-3?/  =  0. 

3.  x-2y  =  l.  6,    'Sx  +  2y  =  12. 
4c.    3x  =  y.  7.    2  2/  -  ar  =  0. 

8.  Draw  the  graph  of  3  a;  =  2  [i.e.,  3x-\-0'y  =  2,  cf .  Ex.  5, 
p.  220];  of  22/  =  5;  of  x'  =  -l;  of  ar  =  9. 

9.  What  is  the  graph  of  x  =  0  ?  of  ?/  =  0  ?  Without  making  a 
drawing,  show  that  the  graph  of  2  x  —  7  y  =  0  must  pass  through 
the  point  0  in  which  the  axes  intersect.  Is  this  true  for  the  graph 
of  every  equation  of  the  form  ax  -\-by  =  0? 

10.  In  the  equation  Ax  — 5  y  =  10,  when  x  =  0,  y=?  When 
y  =  0,  x=?  From  these  two  solutions  of  the  equation  draw  its 
graph  (cf.  §  140,  Note). 

Draw  the  graph  of : 

11.  x-y  =  0.  17.   5x-2y  =  20. 

12.  x-\-y  =  0.  18.   Sx  +  5y  =  7^. 

13.  3  a;  =  —  11.  19.    7x  —  y  =  S^. 

14.  ?/2-16  =  0.  20.   Dx-\-2y=21. 

15.  2x-\-Sy  =  6.  21.   4:X  =  y\ 

16.  2x-3y  =  6.  22.   3a^-4i/  =  0. 

Calling  the  coordinate  axes  S'S  and  TT  instead  of  X'X  and 
Y'  Y,  draw  the  graph  of : 

23.    4:t  =  Ss.  24.    s-t  =  5.  25.    2t-3s^  =  0. 

141.  Drawing  of  graphs  (continued).  Thus  far  we  have 
considered  only  the  simplest  kind  of  graphs ;  the  method 
employed  will  serve,  however,  for  any  equations  whatever 
in  two  unknown  numbers. 


224 


HIGH  SCHOOL   ALGEBRA 


[Ch.  XIII 


Ex.  1.    Construct  the  graph  of  2y  —  x^  =  0. 

CoxsTRucTiON.     Solving  this  equation  for  y  in  terms  of  x,  and 
tabulating  the  corresponding  values  of  x  and  y,  we  obtain : 


X 

y 

Points 

0 

1 

2 

0 
4 

0 

A 
B 

-1 

-2 

-4 

H 
K 

: 

: 

' 

Yi- 

T 

L 

it 

-4^ 

jT 

^                     t             V- 

^^       -    ,-i^    i 

v^ 

iy 

if 

^ 

J 

j 

I 

I 

'     ^ 

•      X 

On  locating  these  points  and  connecting  them  in  order,  we  ob- 
tain the  required  graph,  viz., :  KHOA  •••. 

Ex.  2.    Construct  the  graph  of  4  a^  +  9  2/^  =  144. 

Construction.     Proceeding  as  in  Ex.  1,  we  obtain : 


a;  =  |Vl6- 

2/^. 

y 

0 

X 

Points 

6  or  -6 

^or^' 

1 

1  Vis  or  -  1  \/l5 

BovB' 

-1 

1  Vis  or  -  f  Vl5 

Hov  H' 

-2 

3  V8    or  -  3  \/3 

Kov  K' 

.IjL 

AJ^"--^ 

^^ 

^. 

V    ^/ 

S 

^44- 

0        -2.]"-^ 

4^ 

XaT 

n\ 

t 

#s.± 

n'       ^^ 

\7 

I 

On  locating  these  points,  using  approximate  values  of  the 
square  roots,  and  connecting  them  by  a  smooth  curve,  we  obtain 
the  graph  ABNAN'A. 


141-142 ]      GliAPHIC  liEPRESENTA  TION  OF  EQ  UA  TIONS       225 


Note.  The  limitations  of  the  graph  in  Ex.  2  are  interesting.  Thus,  since 
X  =  |\/16  —  2/2,  X  must  be  imaginary  when  y  is  greater  than  4  ;  hence,  as  our 
graphic  representation  admits  real  values  only,  there  are  no  points  on  the 
curve  whose  ^/-coordinate  is  greater  than  4.  Similarly,  it  may  be  shown  that 
there  are  no  points  on  the  graph  below  y  =  —  4.  And  solving  the  given  equa- 
tion for  y  in  terms  of  x  shows  that  there  are  no  points  on  the  graph  at  the 
right  of  ic  =  6,  or  at  the  left  of  aj  =  —  6. 

Ex.    3.   Construct  the  graph  of  xy  =  4. 
Construction.     Proceeding  as  in  Exs.  1  and  2,  we  obtain: 
4 

X 


X 

y 

Points 

0 

00 

1 

4 

A 

2 

2 

B 

3 

1 

C 

. 

• 

• 

• 

• 

• 

. 

• 

• 

-1 

-4 

H 

-2 

-2 

K 

Y 

1 

1 

\a 

V 

\ 

B 

X 

^, 

N 

'< 

— 

- 

— . 

^ 

o 

X 

N 

K 

\ 

\ 

fl\ 

- 

On  locating   these  points  and  connecting  them  by  a  smooth 
curve,  we  obtain  the  graph  AB  •  •  •  HK. 

142.    Intersection  of  graphs.     Since  (0,  —  3)  and  (4,  0)  are 

solutions  of  the  equation  Z  x  —  ^y 
=  12,  therefore  its  graph  is  the  line 
AB  in  the  figure  (of.  §  140,  Note). 

If  we  now  draw  the  graph  of 
3  rr  +  «/  =  2,  using  the  same  axes  as 
before,  we  obtain  the  line  HK. 

Moreover,  since  P,  the  point  in 
which  AB  and  HK  intersect  (i.e, 
cut)  each  other,  lies  on  each  of  these 


-L4-F- 

kY^ 

r 

X 

^                           B 

,                  >^ 

-_U     ^^     - 

xT        o^H  7^    -x 

W- 

^^ 

^'^  -t 

^^       t 

^^           \h 

v' 

X 

226  IIIGa   SCHOOL   algebra  [Ch.  XIII 

graphs,  therefore  its  coordinates  (§  138)  must  satisfy  each 
of  the  given  equations  (cf.  §  139). 

Hence,  we  may  find  the  coordinates  of  P  by  merely  solving 
the  given  equations  as  in  §  101,  and  without  even  drawing 
their  graphs. 

Approximate  values  of  the  coordinates  of  P  may,  of  course, 
be  found  by  direct  measurement  of  OM  and  MP ;  this 
measurement  constitutes  a  graphical  solution  of  the  given 
equations.  Let  pupils  use  both  methods  for  finding  these 
coordinates,  and  compare  results. 

Remark.  From  what  has  just  been  said,  and  from  the  defini- 
tions in  §  100,  it  follows  that  (let  pupils  explain  why)  : 

(1)  The  graphs  of  consistent  equations  intersect  each  other. 

(2)  The  graphs  of  inconsistent  linear  equations  are  parallel 
lines. 


EXERCISE  XCVII 

Construct  the  graph 

of 

: 

4.    y'  =  Sx. 

8.    xy  =  5. 

5.  y={x-iy. 

9.    3x'  +  4.f-  =  12. 

6.   x'  +  y'  =  25. 

10.    3x'-4.y'  =  12. 

7.   16a^+/  =  64. 

11.    4.f-  =  a^. 

12.  Show  from  the  equation  of  Ex.  4  that  no  part  of  its  graph 
lies  to  the  left  of  the  ^/-axis  (the  line  Y'  Y) . 

13.  Show  from  the  equation  of  Ex.  6  that  no  part  of  its 
graph  lies  outside  a  certain  square  whose  side  is  5 ;  similarly, 
show  that  the  graph  of  Ex.  7  is  contained  within  a  certain  rec- 
tangle whose  dimensions  are  16  and  4. 

14.  Show  from  the  equation  of  Ex.  8  that  its  graph  consists  of 
two  infinitely  long  branches,  one  in  the  quarter  XOY  and  one  in 
the  quarter  X'OY'. 

15.  If  the  graph  of  xy  =  —  5  were  drawn,  how  would  it  differ 
from  that  of  xy  =  5?     Why ? 


142-143J       GEAPHW  REPRESENTATION  OF  EQUATIONS      227 

16.  Draw  the  graph  of  2x-\-y  =  dr -{-S  and  show  that  this 
graph  differs  from  that  of  Ex.  5  only  in  being  moved  two  divi- 
sions upward.     Explain  why  this  should  be  so. 

17.  Find,  both  by  solving  the  equations,  and  by  measurement, 
the  coordinates  of  the  point  in  which  the  graphs  of  x-{-y  =  ^  and 
2  a;  —  2/  =  4  intersect ;  compare  your  results. 

In  each  of  Exs.  18-23  below  find,  as  in  Ex.  17,  the  coordinates 
of  the  point  in  which  the  graphs  of  the  two  equations  intersect : 

18.    i  '  21.    '     ^^         ' 


\2y-x= 


6.  2a;  +  2?/  =  8. 


^^     1 2.^^72/,       .  22^     [^•+2/=3, 


U-4- 
\\x 


y-x=5,    ^  [^x  +  ^y=li. 

2^     {Sy  +  2x  =  17,  ^^     \2x-y  =  S,      (cf.  §§   139, 


2x-y  =  5.  [3y-x'  =  0.         140.) 

24.  How  are  the  graphs  of  two  first  degree  equations  in  a;  and 
y  related  when  the  equations  are  inconsistent  (cf.  Ex.  21)  ? 
when  they  are  simultaneous  and  independent  (cf .  Ex.  20)  ?  simul- 
taneous and  not  independent  (cf.  Ex.  22)  ? 

143.   Graphic  solution  of   simultaneous  equations.     If  the 

graph  of  one  of  two  simultaneous  equations  is  drawn  across 
the  graph  of  the  other  (^i.e.,  if  the  same  axes  are  used  in 
both  drawings),  then  the  measured  coordinates  of  each  point 
in  which  these  graphs  intersect  constitute  an  approximate 
solution  of  the  given  system  (cf.  §  142).  The  following 
examples  will  illustrate  this  procedure. 

Ex.  1.    Solve  graphically  the  simultaneous  equations 
Sx-4:y  =  12, 
3x-^y=2. 

Solution.  The  graphs  of  these  equations  are  the  lines  AB 
and  HK  (figure,  §  142)  ;  and  the  (measured)  coordinates  of  P, 
their  point  of  intersection,  are  approximately  .x-  =  |  and  y  =—2, 
which,  by  trial,  are  found  to  be  a  solution  of  the  given  system. 


228 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XIII 


Ex.  2.   Solve  graphically  the  system  J  ^  ^' 

\y  +  3  =  2x. 

Solution.  The  graphs  of 
these  equations  are,  respec- 
tively, SPOQT  and  AB. 
The  coordinates  of  P,  one  of 
their  points  of  intersection, 
are  approximately  a:  =3.4  and 
y  =  3.7,  which  constitute  an 
approximate  solution  of  the 
given  equations. 

So,  too,  the  coordinates  of 
Q  (viz.,  X  =  .65  and  y  =  —1.6) 
constitute  an  approximate 
solution  of  the  system. 


-y^       y 

Ml               ^- 

l!    ^-^"^ 

2^^ 

^^ 

VI 

4^    ZJ-                           ~t 

X     "--jt                           X- 

^^t 

M 

f\ 

/              ^s. 

^           ^^ 

^                  >-       ~ 

^r 

Ex.  3.    Solve  graphically  the  system 


xy  =  4:. 


Solution.  The  graphs 
of  these  equations  are,  re- 
spectively, AB'A'B  and 
CPQC'P'Q'-,  and  the  co- 
ordinates of  P,  one  of  their 
points  of  intersection,  are 
approximately  x  =  1.4  and 
y=  2.9,  which  constitute  an 
approximate  solution  of 
these  equations. 

So,  too,  the  coordinates 
of  Q  (x=  5.75  and  y  =  .7), 
P'(x  =  -1.4  and  y=  -2.9), 
and  Q'  (x  =  —  5.75  and  y  = 
—  .7)  are  approximate  solutions  of  the  given  equations. 

Note  to  the  Teacher.  In  the  case  of  a  pair  of  simple  equations  the 
solution  by  the  method  of  §  101  is  usually  easier  than  the  graphic  method, 
and  its  results  are  exact  instead  of  approximate.  There  are,  however,  many 
other  cases  in  which  the  graphic  method  is  advantageous  ;  hence  some  prac- 
tice with  it,  even  on  simple  equations,  is  recommended. 


Y  J 

1 

c 

[ 

\ 

\p 

^*-'"*'^ " 

"-k"** 

■^s 

t^ 

Si 

X 

0 

X 

Ar 

^^     "     I 

) 

n 

A        X 

-   ^; 

s^^ 

y 

_'^-^»: 

-^ 

pTi 

5^ 

-    T- 

7- 

14^-144 J      GRAPHIC  liEPRE^EN TA  TION  OF  EQ UA  TIONS      229 

EXERCISE  XCVIll 

Solve  graphically  the  following  systems  of  equations,  and  check 
your  results  as  the  teacher  directs : 

^     {x-^y  =  3,  ^^     ja.'-  +  2/'  =  25, 


5. 


6. 


8. 


10. 


11. 


12. 


x  —  y  =  3. 

12/  =  ^- 

2x-y  =  5, 

14. 

■x'  +  f^  =  2o, 

4.x  =  W-y. 

x-\-y=^l. 

4?/4-3a;  =  5, 

15. 

y  =  3x  +  2, 

0^=5. 

x^  =  4.-y\ 

x  +  y  =  A, 
y  =  2-x, 

5x-10y  =  S6, 
2x-\-3y=-S. 

16. 
17. 

{  .       4.x 

x  =  l. 

xy=-  10, 
x+y  =  2. 

4a.-  +  |2/  =  6, 
ix-^y=S. 

18. 

ix'+9==y, 

\y  =  x^-5x-\-G. 

2x^y\ 

19. 

'x'-^y^  =  25, 

2y=.x. 

[xy=-^. 

2a;-/-l  =  0, 

20. 

'4.x'-9y'  =  S6, 

2a;  +  6v/  +  T  =  0. 

y4-r  =  25. 

x-2y=-12, 

21. 

y  =  Bx—15, 

y-x^=-2x-2. 

x^-9x-  +  23x-15  =  'y. 

By  referring  to  the  graphs  in  the  above  exercises,  find  the 
number  of  solutions  of  a  system  consisting  of: 

22.  Two  simple  equations. 

23.  A  simple  and  a  quadratic  equation. 

24.  Two  quadratic  equations. 

144.  Graphic  solution  of  equations  containing  but  one 
unknown  number.  By  slightly  extending  the  method  of 
§  143,  we  may  find  graphic  solutions  for  quadratic  equations 
in  one  unknown  number. 


230 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XUI 


IT 

-\                  \- 

w          jfe 

M                   -j^ 

r-             ^-4 

X       t 

\         I 

^        f 

"^r    lo  j_  :S 

-^^  m     7^ 

_^    L 

^-4- 

v/      ^ 

F               -^ 

|2/  =  ^= 
12/  =  0. 


Thus,  the  roots  of  x-  —  2x  —  2  =  0  are  manifestly  the  values  of 
X  found  b}^  solving  the  pair  of  simultaneous  equations : 

2x-2,  (1) 

(2) 
Now  the  graph  of  (2),  viz.,  the  a>axis, 
cuts  the  graph  of  (1),  viz.,  the  curve  MQS, 
in  the  points  P  and  E,  whose  coordinates 
are  (approximately)  ic  =  2.75,  y  =  0,  and 
x=  —  .75,  y=0.  And  since  (§  143)  each 
of  these  pairs  of  values  constitutes  an 
approximate  solution  of  (1)  and  (2),  there- 
fore 2.75  and  —  .75  are  approximate  roots 
of  a;2_2,^_2::=0. 


EXERCISE  XCIX 

Find  graphic  solutions  of  : 

1.  x:'-6x  +  S  =  0.  3.    6a;H-5x-4  =  0. 

2.  x''-Sx-\-5  =  0.  4.    a^_3a;2-6a;  +  8  =  0. 

5.  Show  graphically  that  Ax^— 4:X-—llx-\-6  =  0  has  one 
root  between  0  and  1,  and  a  second  root  between  —  1  and  —  2. 
What  is  the  third  root  of  this  equation  ? 

6.  Show  that  one  root  of  cc^  —  7  x-  +  9  .^•  =  1  lies  between  1  and 
2.      Between  what  integers  do  each  of  the  other  two  roots  lie? 

7.  Corresponding  to  any  given  value  of  x,  how  does  the  value 
of  y  \\\  y  =  x'  —  6x-\-6  compare  with  its  value  in  y  =  x^  —  6  x-{-7 ? 
Could,  then,  the  graph  of  the  second  equation  be  obtained  by 
merely  moving  that  of  the  first  upward  through  one  division  ? 

8.  Compare  the  graphs  of  2/=2a7^—10a;— Sand ?/=2ic^—10aj+l; 
also  those  of  y  =  3-\-4:X  —  x''^  and  y  =  10-\-4:X  —  x"\ 

9.  By    first     constructing     the    graphs     of    y  =  oi^  —  6x  +  6, 


y  =  x^  —  6  X  -\-  7,    etc.,    compare   the    roots    of 


X- 


6  a;  +  6  =  0, 


X'  -  6  X -\-7  =  0,  x'  -  6  X  +  S  =  0,  x^ -  6  X -{-9  =  0,  X- -  6x -{-10  =  0, 
anda;2-6a;4-ll  =0. 


144-14.-.]       linAPlllr  UEritESENTATION  OF  EQUATION t>       281 

10.  As  in  Ex.  9  compare  the  two  smaller  roots  of 
ic3  -  7  a;2  +  9  a;  -  1  =  0  with  those  of  a;"^  -  7  a;^  _^  9  ^  _  3  ^  q  ^^^^ 
jB3_7a;2_j_9^_5^() 

Note.  Exercises  9  and  10  illustrate  how,  by  changing  the  absolute 
term  in  an  equation,  a  pair  of  unequal  roots  can  be  made  gradually  to  become 
equal  and  then  imaginary. 

11.  Show  that  the  roots  of  x'^-2x-2  =  0  (§  144)  can  be 
found  from  the  graphic  solution  of  the  system 

y  =  ^,  (1) 

y-2x-2  =  0.  (2) 

12.  Show  that  the  graph  of  Eq.  (1)  in  Ex.  11  may  be  used  in 
the  solution  of  other  quadratic  equations  {e.g.,  x^  -f  5  a?  =  7)  also. 

13.  Is  the  method  given  at  the  top  of  p.  230,  or  that  suggested 
in  Exs.  11  and  12,  to  be  preferred  when  we  have  several  quadratic 
equations  to  solve  graphically  ?    Explain. 

By  the  method  of  Ex.  11  solve : 

14.  a;^-2a^-2  =  0.  17.  x'  =  x-\-^. 

15.  a;"^  — a;  =  0.  18.  12  a;  —  4  a;"' =  5. 

16.  x'  +  x-4.  =  Q.  19.  2a;^-a;  =  -3. 

145.   Use  of  graphs  in  physics,  engineering,  statistics,  etc. 

Descartes's  plan  for  graphically  representing  equations  has 
now  been  adopted  by  practically  all  scientific  men  to  repre- 
sent simultaneous  changes  in  related  quantities.  Physicists, 
chemists,  engineers,  physicians,  statisticians,  etc.,  all  find 
that  this  graphic  representation  of  related  changes  often 
gives  at  a  glance  information  which  could  be  secured  other- 
wise only  by  considerable  effort,  and  that  it  often  brings  out 
facts  of  importance  which  might  otherwise  escape  notice. 

As  a  simple  example  of  the  use  of  graphs  in  this  way  let  us 
consider  the  following  temperature  readings,  taken  from  the 
U.  S.  Weather  Bureau  report,  for  28  hours  beginning  at  noon  on 
Feb.  5,  1906,  at  Ithaca,  N.Y. 


^32 


UIGII  SCHOOL  ALGEnnA 


[Ch.  xm 


IIR. 

Tem. 

- 

-11 

^°H 

- 

T 

■V, 

s 

12 

1 
2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

4 

10° 

9 

8.5 

7 

6 

5 

3.5 

3 

2 

1 

0.5 

0 
-5 
-6 
-7 
-6 
-6.5 
-6 
-8 
-9 
-11 

-3 
0 
2 
3 

3.5 
5 
3 

Si 

s 

— 

3 

=1 
s 

\ 

^i 

\ 

\ 

Z^ 

V 

"  z:^ 

\ 

z 

-Ji 

^, 

\ 

I        i£ 

^O   1   1 

HOURS                                          l_ 

-        ^ 

O    i 

4 

6 

8 

JO  \  1;2  1 

2       4        6        8       10/ 

12      2       4 

\ 

-                 T- 

\ 

r 

\ 

-              ^  - 

_^o 

\ 

7 

\ 

-^^^         /   - 

/ 

sy^'^s,      1    _ 

5     J 

:        si: 

-10° 

\t 

yz     . 

/ 

1 

the 
mi 
hi^ 

tal 
sti 

Her 

3sa 
ich, 
;hes 
^Ids 
nila 
idie 

e 

Ql 

a 
,t 

t 
Lt( 
d 

t 

e 
.n 

a 
h 
id 

a 

d 
n 
is 
\ 
nc 

}    t 

les 
he 

a  ] 

ir 
igi 
Ic 

ab 

tic 

)W 

o^^ 
ifo 
ire 
om 

ul 
)n 
r 

re 
rn 
s 
P 

at 

s 

H 

st 
aa 

y 

ai 

ed  fig 
as  to  i 
)idly  ] 
point 
Ltion  £ 
iekl  i 
ed. 

'ures  and  the  graph  answer 
:he  temperature  —  when,  how 
t  rose  or  fell,  what  were  its 
s,  etc.     The  graph,  however, 
it   a  mere  glance,  while  the 
b  only  after  they  have  been 

Note.  For  other  interesting  applications  of  graphs  see  Tanner  and 
Allen's  Analytic  Geometry,  pp.  73-78.  Also,  and  especially,  "  Graphic  Meth- 
ods in  Elementary  Algebra,"  by  Prof.  William  Eetz,  in  /School  Science  and 
Mathematics,  vol.  6,  pp.  683-687.  This  article  gives  many  good  suggestions 
as  well  as  valuable  material  and  references. 


EXERCISE  C 

By  reference  to  the  above  temperature  graph  (usually  called 
thermograph)^  answer  the  following  questions : 

1.  Between  what  hours  was  the  temperature  below  0°  ?    When 
was  it  lowest  ? 

2.  When  was  the  temperature  falling  most  rapidly  ?     Explain. 


145 J  GRAPHIC  liEPBElsENTATlON  OF  EQUATION H         233 

The  following  tables  give  the  population  (in  millions)  of  the 
countries  named,  for  certain  years  between  1800  and  1900. 


British  Isles 

Lands  now  in  the 
German  Empire 

France 

Unitei 

States 

Year 

Population 

Year 

Population 

Year 

Population 

Year 

Population 

(millions) 

(millions) 

(millions) 

(millions) 

1801 

15.9 

1816 

24.8 

1801 

27.3 

1810 

7.2 

1811 

17.9 

1837 

31.5 

1821 

30.4 

1820 

9.6 

1821 

20.9 

1847 

34.7 

1841 

34.2 

1830 

12.8 

1831 

24 

1856 

36.1 

1861 

37.3 

1840 

17 

1841 

26.7 

1865 

39.4 

1866 

38 

1850 

23.2 

1851 

27.8 

1872 

41 

1872 

36.1 

1860 

31.4 

1861 

28.9 

1876 

42.7 

1876 

36.9 

1870 

38.5 

1871 

31.4 

1885 

46.8 

1881 

37.6 

1880 

50.1 

1881 

34.9 

1895 

52.2 

1891 

38.3 

1890 

62.6 

1891 

37.7 

1896 

38.5 

3.  Taking  the  number  of  years  after  1800  as  the  cc-coordinate 
and  the  population  (in  millions)  as  the  ^/-coordinate,  locate  the 
several  points  represented  by  the  above  table  for  the  British  Isles, 
and  join  these  points  by  straight  lines.  Similarly,  draw  graphs 
for  the  remaining  tables. 

4.  By  reference  to  your  graphs,  compare  the  population  of  the 
countries  named  in  1830;  in  1856;  in  1880.  By  reference  to  the 
tables  compare  the  populations  in  1871. 

5.  In  making  comparisons  like  those  of  Ex.  4,  is  it  easier  to 
use  the  tables  or  to  use  the  graphs  ?    Why  (cf.  §  145)? 

6.  By  reference  to  the  graphs  answer  the  following  questions  : 

(1)  When  did  the  population  of  the  United  States  first  exceed 
that  of  the  British  Isles  ?  that  of  France  ?  that  of  Germany  ? 

(2)  During  what  years  has  the  population  of  the  United  States 
increased  most  rapidly  ? 


CHAPTER   XIV 
IRRATIONAL  NUMBERS  -  RADICALS 

146.  Preliminary  remarks  and  definitions.  A  number 
which  may  be  expressed  as  the  quotient  of  two  integers, 
positive  or  negative,  is  called  a  rational  number. 

J^.^.,  3(  =  f);  -7l(  =  ^);  2. 75(  =  |f|),  etc.,  are  rational 
numbers. 

Nearly  all  the  numbers  thus  far  used  have  been  rational, 
although  we  have  met  a  few  such  forms  as  V2  and  V  — 5. 

In  this  and  the  next  chapter  we  shall  examine  more 
closely  such  numbers  as  V2  and  V—  5.  These  numbers  are 
particular  cases  of  a/^,  which  is  defined  (§  113)  by  the 
equation  (VaY^a;  hence  ( V2)2  =  2  and  (V^=~5)2=-5. 

The  numbers  V2  and  V  —  5  resemble  each  other  in  that 
neither  of  them  is  rational  (since  no  rational  number  squared 
is  2  or  —  5),  but,  as  we  shall  soon  see,  they  differ  widely 
in  another  regard. 

By  squaring  1  and  2,  we  find  that  V2  is  greater  than  1  and 
less  than  2  ;  then  by  squaring  1.1,  1.2,  1.3,  •••,  we  find  that 
V2  is  greater  than  1.4  and  less  than  1.5;  similarly,  V2  is 
greater  than  1.41  and  less  than  1.42  ;  etc. 

Since  V2  lies  between  1  and  2,  1.4  and  1.5,  1.41  and  1.42, 
etc.,  therefore,  if  1.4  (or  1.5)  is  taken  for  V2,  the  error  is 
less  than  0.1 ;  if  1.41  (or  1.42)  is  taken,  the  error  is  less  than 
0.01;  and,  by  continuing  this  process,  we  can  find  rational 
numbers  which  approximate  V2  to  any  required  degree  of 
accuracy. 

234 


140-147]  IRRA  TIONAL  NVMBEIiS  —  liADTCA LS  235 

On  the  other  hand,  since  the  square  of  any  rational  num- 
ber is  positive,  therefore  we  cannot  express  V  —  5,  even 
approximately,  by  means  of  rational  numbers. 

Numbers  like  V2,  which  are  not  rational,  but  which  may 
be  expressed  approximately  to  any  required  degree  of  accuracy 
by  means  of  rational  numbers,  are  called  irrational  numbers. 

^.g.,  V2,  5  —  V7,  and  10  4- v2  are  irrational  numbers. 

Numbers  like  V  —  5,  wliich  cannot  be  expressed,  even 
approximately,  by  means  of  rational  numbers,  are  called 
imaginary  numbers  (cf.  §  114,  Note).  Rational  and  irra- 
tional numbers  taken  together  are  called  real  numbers. 

Note  to  the  Teacher.  Emphasis  should  be  laid  upon  the  fact  that 
although  such  numbers  as  V2  can  be  expressed  only  approximately  by  means 
of  rational  numbers,  they  are,  nevertheless,  just  as  exact  and  definite  as  are 
integers  and  fractions.  n 

Thus,  let  ABCD  be  a  square  whose  side  AB  is  1  foot  long, 
and  let  x  represent  the  number  of  feet  in  its  diagonal  AC, 
then  it  is  easily  proved  by  geometry  that 
cc2  =  2,  i.e.,  that  x=y/2. 

The  numbers  1, 1.4, 1.41, 1.414, 1.4142,  etc.,  are  successive 
approximations  to  the  length  of  this  diagonal,  but  its  exact 
length  is  V2 ;  hence  the  necessity  of  including  such  numbers  as  y/2,  in  our 
number  system. 

It  will  be  worth  while  also  to  connect  this  latest  extension  with  the  exten- 
sions previously  made  (see  p.  16,  footnote).  Thus  fractions  arose  from  gen- 
eralizing division  ;  negative  numbers  arose  from  generalizing  subtraction  ;  and 
in  the  present  article  it  appears  that  generalizing  evolution  introduces  two 
further  new  kinds  of  numbers,  viz.,  the  irrational  and  the  imaginary. 

In  other  words  :  while  the  direct  operations  (viz.,  addition,  multiplication, 
and  involution)  with  positive  integers  always  produce  results  that  are  positive 
integers,  the  inverse  operations  (viz.,  subtraction,  division,  and  evolution) 
lead  respectively  to  negative,  fractional,  and  irrational  and  imaginary  num- 
bers, and  demand  for  their  accommodation  that  the  primitive  idea  of  number 
be  so  enlarged  as  to  include  these  new  kinds  of  numbers. 

147.  Further  definitions.  An  indicated  root  is  usually 
called  a  radical;  the  number  whose  root  is  indicated  is 
called  the  radicand.  If  the  root  is  irrational,  but  the  radi- 
cand  rational,  the  expression  is  often  called  a  surd.     Thus, 

HIGH   SCH.    ALG. — 16 


236  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 


V2,  a/8,  6-v/45,  V  —  2,  and  V5  +  VlO  are  radicals,  whose 
respective  radicands  are  2,  8,  etc. ;  of  these  radicals  V2  and 
6V45  alone  are  called  surds. 

The  coefficient  of  a  radical  is  the  factor  which  multiplies 
it,  and  the  order  of  the  radical  is  determined  by  the  root 
index.  Two  radicals  which  have  the  same  root  index  are 
said  to  be  of  the  same  order.  Thus,  the  surds  12V5  ax^  and 
m2V674  are  of  the  same  order,  viz.,  the  7th,  and  their  co- 
efficients are  12  and  m^  respectively. 

Surds  of  the  second  and  third  orders  are  usually  called 
quadratic  and  cubic  surds,  respectively. 

Radicals  which,  when  simplified,  are  of  the  same  order 
and  have  their  radicands  exactly  alike  are  called  similar 
(also  like)  radicals  ;  otherwise  they  are  dissimilar  (unlike). 
Expressions  which  involve  radicals,  in  any  way  whatever, 
are  called  radical  expressions ;  they  are  monomial,  binomial, 
etc.  (cf.  §  20),  depending  upon  the  number  of  their  terms. 
Thus,  V5  and  3V5  are  similar,  monomial,  quadratic  surds, 
while  b  a  -\-  3  V7  and  2  V9  +  3  Va:  are  binomial  surds. 

148.  Principal  roots.  We  have  already  seen  that  a  number 
has  two  square  roots  (e.^.,  V9  is  +  3  or  —  3),  and  we  shall 
see  later  that  every  number  has  three  cube  roots,  four  fourth 
Yoot^^  five  fifth  roots,  etc. 

E.g.,  a/8  =  2,  -1+V":r3,  or  - 1 -V^^,  since  the 
cube  of  each  of  these  numbers  is  8 ;  and  VI6  =  2,  —  2,  2V— 1, 
or  -  2V^^. 

Although  the  number  of  roots  always  equals  the  order  of 
the  radical,  not  more  than  two  of  these  roots  can  be  real;  and 
when  there  are  two  real  roots,  they  are  opposite  numbers. 
By  the  principal  root  of  a  number  is  meant  its  real  root.,  if 
there  is  but  one  real  root,  and  its  real  positive  root  if  there 
are  two  real  roots. 

E.g..,  if  attention  is  confined  to  principal  roots,  V9  =  3 
(and  not  -  3),  -V^^  =  -  2,  ^125  =  5,  a/T6  =  2,  etc. 


147-148]  IRRATIONAL  NUMBERS  —  RADICALS  237 

EXERCISE   CI 

1.  What  is  a  rational  number  ?  Use  your  answer  to  show- 
that  7,  f ,  —  8|-,  and  V36  are  all  rational. 

2.  What  is  an  irrational  number  ?  Is  ^/8  an  irrational  num- 
ber ?     Why  ? 

3.  By  the  method  used  in  §  146  for  V2,  find  two  approximate 
values  for  V3  (one  larger  and  the  other  smaller  than  the  true 
value)  which  differ  from  V3  by  less  than  0.001. 

4.  Find  two  successive  approximations  to  the  value  of  V5. 
Compare  these  approximations  with  the  result  of  extracting  the 
square  root  of  5  by  the  method  of  §  118. 

5.  What  is  an  imaginary  number  ?  Give  several  illustrations. 
For  what  values  of  n  is-v^'—  5  imaginary  ? 

6.  Is  the  number  2l4-Vl7  rational  or  irrational?  Why? 
What  kind  of  number  is  84 V5 -  ^^-^ ?     Why  ? 


7.  Are  both  ^'21  and  v  2  +  V7  radicals  ?     Are  they  surds  ? 
Are  all  radicals  surds  ?     Are  all  surds  radicals  ? 

8.  In  Exs.  44-54,  p.  242,  point  out  the  coefficient  of  each  surd. 
May  the  coefficient  of  a  surd  be  fractional  ?  negative  ? 

9.  Write  a  surd  of  each  of   the  following  orders :    2d,    5th, 
3d,  7th. 

10.  Define  similar  surds,  and  illustrate  your  definition.  May 
the  coefficients  differ  and  the  surds  still  be  similar  ? 

11.  What  factor  do  two  similar  surds  necessarily  have  in  com- 
mon? What  kind  of  a  number,  then,  is  the  quotient  of  two 
similar  surds  ?     Illustrate  your  answer. 

12.  Write  a  monomial  cubic  surd ;  a  binomial  quadratic  surd ; 
a  trinomial  surd  of  the  5th  order. 

13.  How  many  values  has  Vl6?  What  are  they?  What  is 
the  principal  square  root  of  16  ?  What  is  the  principal  fifth  root 
of  —  32  ?     Define  the  principal  root  of  a  number. 


238  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

149.  Principles   involved  in  operations  with   radicals.     If 

we  exclude  imaginary  numbers,  the  principles  employed 
in  operations  with  radicals  may  be  symbolically  stated 
thus: 

(i)  Vlcy  =  ^Hc  •  Vy^ 

(ii)  4-=—^ 

(iii)  'Vx  =  ^'Vx  =  ^x, 

wherein  n  and  t  are  positive  integers,  while  x  and  y  may 
have  any  values  whatever,  except  that  they  cannot  be  nega- 
tive when  the  root  index  is  even. 

150.  Proof  of  the  principles  in  §  149.  For  the  sake  of  sim- 
plicity, we  shall  (1)  limit  the  proofs  to  principal  roots,  and 
(2)  assume  that  a  change  in  the  order  of  the  factors  of  a 
product,  even  when  these  factors  are  irrational  numbers, 
leaves  the  product  unchanged  (cf.  §  42). 

With  these  restrictions  the  correctness  of  (i),  (ii),  and 
(iii),  §  149,  follows  from  the  meaning  of  the  symbol  Va 
(§  113).     Thus,  to  prove  (i)  we  proceed  as  follows  : 

Qy/xVyY  =  VxVy  •  Vx  Vy-  •••to  n  factors  [§  9 

=  xy ;  [since  (  Va)"  =  a 

whence     Vx^y^^'xy,  [§113 

which  was  to  be  proved. 

This  principle  may  be  translated  into  words  thus : 

The  nth  root  of  the  product  of  two  numbers  equals  the  product 
of  the  nth  roots  of  these  numbers. 

The  proofs  of  (ii)  and  (iii)  are  left  as  an  exercise  for  the 
pupil. 


149-101]  IHRATIONAL  NUMBERS  —  liADICALS  239 

EXERCISE  Cll 

Verify  the  following  equations : 

1.  V9- V25  =  V9.25.  3.   Vi6T9  =  Vl6.  V9. 

2.  ^/38.^27=^-8.27.   4.   ■v'lOOO  a«  =  ^125  a'  •  -y/S. 

5.  Show  that  Exs.  1-4  are  special  cases  of  §  149  (i). 

6.  Find  V5  •  v3  correct  to  two  decimal  places  (§  118)  ;  then 
find  Vl5  (i.e.,  VS  •  3)  correct  to  two  decimal  places,  and  compare 
results.  Does  this  exercise  illustrate  any  jwactical  advantage  in 
knowing  that  -\/x  •  -s/y  =  -Vxy  ?     Explain. 

7.  By  means  of  §  149  (ii),  show  that  V35-^V7=V5,  and 
that  VlOa^-^-  V—  2a  =  V  — 8a.     State  §  149  (ii)  in  words. 

8.  Find  (correct  to  two  decimal  places,  §  118)  V7  -^  V5,  also 
Vl.4  (i.e.,  VT^S),   and   compare  results.     Does   this   exercise 

n/  ni 

illustrate  any  practical  advantage  in  knowing  that  ^^  =  -y-  ? 

-\/y      ^y 

9.  Show   that   f^X^^    and   thus   prove  that  V*/^-^. 

Verify  that  "Vx  =  yj</x  =  ^  VS  [cf.  §  149  (iii)]  when 

10.  n  =  2,    p  =  2,     a;  =  81.         12.    n  =  4,    p  =  3,     a;  =  c-*d^l 

11.  n  =  5,    p  =  2,     x  =  m^.        13.    n  =  2,    p=^S,     a;  =  64. 

14.  Use  §  149  (iii)  to  find  -^49  (i.e.,  \/49),  correct  to  two 
decimal  places ;  also  find  Vl44.  How  may  you  find  the  9th 
root  of  any  given  number?  the  8th  root? 

15.  Prove  the  correctness  of  §  149  (iii).  What  do  n  and  t 
represent  in  this  principle  ?      May  nt,  then,  equal  11  ?     Explain. 

151.  Special  cases  of  §  149.  Besides  the  three  principles 
given  in  §  149,  these  four  others  are  often  useful  : 

(i)  V^  =  jrV/, 

(ii)  -^jr^...  =Vx-V/-  Vz--, 

(iii)  ^^=(^xy, 

(iv)  V]r=A/i^. 


240  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

The  correctness  of  these  principles  may  be  established  by 
the  method  used  for  the  proof  of  (i),  §  149 ;  it  is  easier,  how- 
ever, to  regard  them  as  special  cases  of  §  149,   thus: 

■v/i^  =  V^V^,  [§149(i) 

which  establishes  (i)  above. 

So,  too,  ^xyz  •••  =  Vx  '  ^yz  •••  [§  149  (i) 

=  ^x  '  -\/y  '  -Vz  •••  =  etc., 
which  proves  (ii)  above  ;  and  (iii)  follows  from  (ii)  by  let- 
ting x  =  y  =  z  =  '•',  and  supposing  the  number  of  these  fac- 
tors to  be  t. 

Again,  %/^=a/^  [§  149  (iii) 

,  .  ,  ,.   .  =V:c%  [since  </(xy  =  x'- 

which  proves  (iv).  *-  ^    ^ 

EXERCISE  cm 

1.  Is  SVE  equal  to  VS^Ts?  Why?  May  2-^  be  written 
as  V2^  •  6  ?  Why  ?  How  may  the  coefficient  of  a  radical  be 
inserted  under  the  radical  sign  [cf.  §  151  (i)]? 

2.  By  means  of  §  151  (i)  show  that  V20  =  2V5,  and  also  that 
V— 54=— 3V2.  How  may  we  simplify  a  square  root  which 
contains  a  square  factor?  a  5th  root  which  contains  a  factor 
raised  to  the  5th  power  ? 

3.  By  the  method  of  §  150,  show  that  x^y  =  V^. 

4.  Verify  that  V2  •  V3  •  V5  =  V30 ;  find  at  least  two  deci- 
mal places  in  each  member  of  the  equation  (cf.  Ex.  6,  p.  239). 
How  find  the  product  of  several  radicals  of  the  same  order  ? 

5.  By  the  method  of  §  150,  show  that  ■\/x  •  -Vy  •  -\/z  =  -Vxyz. 

6.  Find,  correct  to  two  decimal  places,  (V6)^,  also  V6^,  and 
compare  results.  How  would  you  raise  Vl5  to  the  second 
power?  to  the  5th  power?     Explain. 

7.  Show,  using  the  method  of  §  150,  that  (^x)  =  -y/af. 


151-152]  IRRATIONAL  NUMBERS  —  RADICALS  241 

8.  Verify  that  ■\/aP  =  -^a^,  and  that  -\/a^^  =  Va\  In  each 
of  these  equations  compare  the  exponents  of  a,  also  the  root- 
indices. 

9.  Is  a  radical  changed  in  value  if  we  multiply  its  root- 
index  and  also  the  exponent  of  its  radicand  by  the  same  factor 
[cf.  §  151  (iv)]?  if  we  divide  both  by  any  factor  common  to 
them  ?     Illustrate. 

10.  Using  §  151  (iv),  show  that  -^9  (i.e./^3^)  =  V3  =  ^8i. 
Which  is  more  easily  computed,  V32,  or  its  equal,  V2? 

152.   Reduction  of   radicals  to  their  simplest  forms.      A 

radical  is  said  to  be  in  its  simplest  form  when  the  radicand 
is  integral,  when  the  index  of  the  root  is  as  small  as  possible, 
and  when  no  factor  of  the  radicand  is  a  perfect  power  corre- 
sponding in  degree  with  the  indicated  root. 

The  following  examples  illustrate  the  application  of  the 
foregoing  principles  in  the  reduction  of  radicals  to  their 
simplest  form. 

Ex.  1.   Reduce  VSoV  to  its  simplest  form. 


Solution.  V8  aV  =  V4  a V  .  V2  ax  [§  149  (i) 

=  2  ax^  V2  ax. 
Ex.  2.    Reduce  V4  a^x^y^  to  its  simplest  form. 

Solution.         -s/A.o'xY  =  </{2  aa^i/f  =  ^2^^.       [§  151  (iv) 
Ex.  3.    Reduce  Vf  to  its  simplest  form. 

Solution.  -^  -  =  -J  — 1 


5_2 
5.52 


=^^\>m=Ujm,  [§i5i(i) 


EXERCISE  CIV 
Reduce  each  of  the  following  to  its  simplest  form : 
4.    V18  (^.e.,  V9^).     6.    V45.  8.    S^IG. 

5      V24.  7.    V75.  9.    S^^^^. 


242 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XIV 


29. 


45 


/27  a? 


10.  2\/54. 

11.  ^32. 

12.  VI  0-.e.,ViT7).    30.    Va^  (cf.  Ex.  2).    46.   i^f. 

13.  VS 

15.  ^^  ^ 

16.  VJ  (cf.  Ex.  3).      34.    V216. 

17.  vi- 


31.  V^ 

32.  Va;y. 

33.  a/25. 


47.  3^125  a^a^. 

48.  y\^±y. 
^  x-y 


18.  lOVf 

19.  </l 

20.  2V|. 

21.  V^^ 

22.  v=i: 


35.  V32m^7i'«. 

36.  3^36  h-x'\ 


49.    ■>/a2"a;"+\ 
50. 


64  m« 


37.    V  32  a VI 
39. 


125 

51.  6^320. 

52.  V- 486^2^ 

53.  |V2|. 


23.  V27a;l 

24.  VSw?. 


\9^ 


54.    Vl8a-9. 


25.  Va«6^'. 

26.  VV^/*. 


40. 


41. 


yV 


55.    Var+Wy^\ 


11, 
2x 


56.    Va2''ft.""+^ 


4  25a« 


57.    </ -  40  a^^-^y\ 


27. 


r 


58.   4  Va^"  —  a*"aJ". 
59. 


28.     J^ 


V 


L    3a^^ 


-2  6a; 


2a 


60.    V3a;2— 6«2/+32/^. 


42.  V162. 

43.  a/36. 

44.  2V|V. 

In   the   following,   insert   the   coefficients   under    the    radical 
signs : 

66.    -4^1. 


61.  3V7. 

62.  5^4. 

63.  2a/6. 

64.  -2  Vs. 

65.  y\V40, 


70.    (c4-l)V5c. 


67.    iVlW-. 

71. 

72. 
73. 

1     , 

O/.       ^    V  X24. 

-VaMo^-i). 

68.    -V72a^l 

4  w3vV4  wv^ 

69-   ~^12a'x. 

-2y'-2^<Jyh\ 

152-153]  IRRATIONAL  NUMBERS  —  RADICALS  243 

153.  Addition  and  subtraction  of  radicals.  Similar  radicals 
(§  147)  may  evidently  be  added  and  subtracted  by  regarding 
the  common  radical  factor  as  the  unit  of  addition.  The  sum 
or  difference  of  dissimilar  radicals  can,  of  course,  only  be 
indicated,  and  this  is  done  by  connecting  the  radicals  with 
the  proper  signs  (cf.  §  23). 

The  radicals  to  be  added  or  subtracted  should  first  be 
reduced  to  their  simplest  forms  ;  the  following  examples 
will  illustrate  the  procedure. 

Ex.  1.     Find  the  sum  of  V75  and  3  Vl2. 

Solution.  Since  V75  =  V25T3  =  ^25  •  V3  =  5  V3,  [§  149  (i) 
and  3  Vl2  =  3  V473  =  3  V4  .  V3  =  6  V3,  [§  149  (i) 

therefore  V75  +  3  Vr2  =  11 V3. 

Ex.  2.    Find  the  sum  of  5  Vl8  and  —  VOX 

Solution.    Since  5  Vl8  =  5  V9^  =5-3  V2  =  15  V2, 

and  -Vo:5^-VT=:--v/-^  =  -|V2, 

therefore  5  Vl8  +  (  -  Vo:5)  =  (15  - 1)  V2  =  14i  V2. 


Ex.  3.   Find  the  sum  of  V9  x  -  18,  6  V4  a;  +  8,  V36  x  -  72,  and 

-V25a;  +  50. 


Solution.     V9  a;  -  18  +  6  V4a;4-8  +  V36a;-72  -  V25a;-f  50 

=  3  V^^^  + 12  V^T2  +  6  V^^^  -  5  a/^T2 
=9  Vx'^  +  7  V^T2^ 

EXERCISE  CV 
Find  the  sum  of  : 

4.  V5,  7  V5,  and  -  3  V5.  8.   2  V20,  -^  V45,  and  Vsa 

5.  Vl8,  V50,  and  V98.  9.    ^250,  -v^,  and  -^U. 

6.  VT2,  V75,  and  V27.  10.    \/500,  -y/WS,  and  ^^^^32: 


7.    V28,  -  2  V63,  and  V700.        11.    <^cdf*,  3  -\/cdf,  V  -  8  c'df. 


244  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

12.  Find  (correct  to  two  decimal  places)  the  value  of  each  of 
the  surds  in  Ex.  7  (cf.  §  118)  and  add  your  results.  How  does 
this  sum  compare  with  that  previously  found  for  Ex.  7  ?  Is  there, 
then,  any  practical  advantage  in  simplifying  surds  before  adding 
them? 

13.  Find  the  sum  of  a,  2  b,  and  c ;  oi  S  x,  4:  y,  2  x,  and  —5y. 

14.  What  isjhe  sum  of  3V2  and  5^/7?  of  3  V2,  5^/7] 
-  2  V7,  and  -V2? 

15.  Write  a  rule  for  the  addition  and  subtraction  of  radicals, 
providing  both  for  those  cases  in  which  the  given  radicals  are 
similar  and  for  those  in  which  they  are  dissimilar. 

Simplify  the  following  expressions  as  far  as  possible  (cf .  §  152), 
and  explain  your  work  in  each  case : 

16.  ^/i35  +  -^625-■^/320.  22.    -^128^+ a/375^- -v^Sl^ 

17.  -^40 +  V28 4- V175 +^25:  /«;,/«_    ja 

19.  V5  +  V75-V12  +  2V3.  4»_    jW^        /£^ 

20.  Vl47-Vi  +  iv/3+J^9:       •    ^bY      \    bf    ^^bf' 

21.  6^|o+4^/I|-8^/|||:      25.    V(a4-6)^c-^(a+6)V. 

26.  ^l92^-2-^3^-^/5^+v40^. 

27.  -y/abx -\- -^^^m^  -  \^S~aW^. 


28.    V3a;3  +  30ar^  +  75a5-V3a^-6a^  +  3a;. 


29.  V5a^  +  30a*  +  45a3-V5a^-40a^  +  80a^ 

30.  V50  +  -V/9-4  VJ  +  -v/24  +  ^27-a/64. 

31.  V|  +  6V|-iVl8  +  ^36-^+^125-VS. 


32.    Va^  —  a^x  —  Vaa??  —  a^  —  V(a  +  a;)(a^  —  a?^. 

154.   Reduction  of  radicals  to  the  same  order.     By  (iv)  of 
§151, 

^5  =  ^-^5^  =^5^  =^625, 
and  ^7  =  ^^P  =  -^^'P=-^^3"43, 


153-154]  IRRATIONAL   NUMBERS  —  RADICALS  245 

i.e.,  the  radicals  V5  and  -^/l  are  equivalent,  respectively,  to 
^625  and  v'348 ;  and  these  last  two  radicals  are  of  the 
same  order,  viz.,  the  twelfth. 

Moreover,  it  is  evident  that,  by  a  similar  application  of 
§151  (iv),  any  two  or  more  radicals  whatever  may  be 
reduced  to  equivalent  radicals  of  the  same  order.  This  new 
order  must,  of  course,  be  some  common  multiple  (preferably 
the  L.  C.  M.)  of  the  orders  of  the  given  radicals. 

EXERCISE  cvi 

1.  Is  -^^ equal  to  Vx?  Is  ^3aV equal  to  ^9aV?  Using 
the  method  of  §  150,  prove  the  correctness  of  your  answer  to  each 
of  these  questions.     Compare  also  §  151  (iv). 

2.  Eeduce  ■y/25  mV  and  a/8  a^6V  to  equivalent  radicals  of 
the  4th  order,  and  explain  (cf.  Exs.  8-10,  p.  241). 

Reduce  to  equivalent  radicals  of  the  order  indicated,  and  ex- 
plain your  work  : 

3.  -s/ay^,  9th  order.  7.   3  ax,  4th  order. 

4.  -^SM,  6th  order.  8.    ^2  m\  12th  order. 

5.  -^2sf,  10th  order.  9.    ^AV,  12th  order. 


6.    V2  sf,  10th  order.  10.    VciVa^,  8th  order. 

11.    Express  as  equivalent  radicals  of  the   6th  order:    -\/' 


m, 


■\/9  m^n\  and  mn. 

Eeduce  the  following  to  equivalent  radicals  of  the  same  order: 

12.  V5  and  ^/li.  19.    Vl4,  ^25,  and  ^95. 

13.  -v/T  and  V3.  20.   3V3,  5V2,  and  2-\/20. 

14.  V27and-\/3^l  21.   2  V3,  3^2,  and  2-v/39. 

15.  V6and3.  22.  3Va&,  a^2^,  and -^^^^ 

16.  -Vp,  V5j^  and  -y/f^.  23.   x'</2^,  V^,and2^^^ 

17.  i/W,  V2,  and  ■^.  24.    </x,  5  xVy,  and  V^. 

18.  ~^2¥y,  \^3a?  and  xy.  25.    ^d^  +  6"^,  and  aVa  ~  b. 


246  inan  school  algebra  [Ch.  xiv 

26.  Can  the  radicals  in  Ex.  22  be  reduced,  to  equivalent  radi- 
cals of  the  6th  order  ?  of  the  12th  order  ?  of  the  9th  order  ? 
Give  the  reasons  for  your  answer  in  each  case. 

27.  What  is  the  lowest  common  order  to  which  you  can  reduce 
the  radicals  in  Ex.  23  ?    those  in  Ex.  24  ?    in  Ex.  25  ? 

28.  By  first  reducing  Vl5  and  V6  to  the  same  order,  show 
that  -\/W  is  greater  than  V6» 

29.  Which  is  greater,  V5  or  Vl2  ?  Explain  your  answer 
(cf.  Ex.  28).  _ 

30.  Which  is  greater,  3  VlO  or  2  a/100  ? 

Hint.    First  insert  coefficients  under  radical  signs  (cf.  Ex.  1,  p.  240). 

Arrange  the  following  radicals  in  order  of  magnitude : 

31.  3  V3,  5  V2,  and  2  v'20.  32.   2  ^39,  3  -y/2,  and  2  V3. 

155.  Product  of  monomial  radicals.  If  two  or  more  radi- 
cals are  of  the  same  order,  their  product  may  be  written 
down  immediately  by  §  151  (ii)  ;  while  if  they  are  of  differ- 
ent orders,  they  should  first  be  reduced  to  equivalent  radi- 
cals of  the  same  order  (§  154). 

Ex.  1.   Multiply  a/5  by  V2^ 

Solution.  The  L.  C.  M.  of  the  orders  of  the  given  radicals  is 
6,  and  by  §154  ^=V5^, 

and  V2=-v/2^; 

therefore  </5'  V^  =  </¥' ''^¥  =  y/W^'  [§  149  (i) 

=  a/200. 

Ex.  2.  Find  the  product  of  4  h-\/ax  and  V5  a^,  and  simplify 
the  result. 

Solution.     As  in  Ex.  1,  4  h^ax  =  4  6  VaV, 

and  -\/5  a^  =  V5^  a^ ; 

therefore  4  b^ax  •  V5  a^  =  4  ft  Va V .  V 5^a* 

=  4  6^52^V  [§  149  (i) 

=  4a6A^25^^.  [§151(i) 


154-156]  IRRATIONAL  NUMBERS  —  RADICALS  '247 


EXERCISE  evil 
Find  the  following  products,  and  simplify  the  results: 

4.  2Vi5  .  V5.  '^2y^y'    

6.   -^  .  ^^.  11.   (8  -  2  Vl5)  .  2  V6. 

13.  How  may  we  find  the  product  of  two  or  more  radicals 
which  are  of  the  same  order? 

14.  How  may  we  find  the  product  of  two  or  more  radicals  of 
different  orders  ?     Illustrate,  using  the  surds  V3  and  V2. 

Find  the  following  products,  and  simplify  the  results : 

15.  -Vab  •  V&c. 


16.  V3  .  3^. 

17.  V2  .  </I. 

18.  2\/5  .  7-^10.  26.   V^'  •  Vl2^  .  V75^^ 


23. 

^f. 

</2f. 

24. 

2V2 

■  ^256. 

25. 

V2- 

^1-^. 

19.   a/i  .  ^(if.  27.  V2a&  .  ^abc  •  ^4:a^b\ 


20.    \/^  .  aP^ 


3/]^    J/W  28.   V9(c-A;)=^.-lV3(c-A;). 

29.  (V72+^12-3  V98)  •  V2. 


21.  ^4a2  .  WSa^ 

22.  ^^  .  V^. 


156.  Multiplication  of  polynomials  containing  radicals.   The 

product  of  two  polynomials  containing  radicals  is  obtained 
by  multiplying  each  term  of  the  multiplicand  by  each  term 
of  the  multiplier  and  adding  the  partial  products,  just  as  in 
the  case  of  rational  polynomials. 


248  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

Ex.  1.   Multiply  5V2-2V3  by  3V2  +  4V3. 

Solution.  5  V2  —  2  V3 

3  V2  +  4  V3 
30  -  Q^Q_ 

+  20V6-24 
30  +  14  V6  -  24  =  6  + 14  V6. 

Ex.  2.  Expand  (2V3— ■v2)^  by  the  binomial  theorem. 

Solution.     (2V3-a/2)2==(2V3)2-2(2V'3)^2+(a/2)2 

=  12-4^108  +  ^4. 


EXERCISE  CVIII 

Multiply,  and  express  your  results  in  the  simplest  form : 

3.    V5  —  5  by  V5  + 1.  fa       S 


/a       4b\  ,      fa       4/6\ 


4.  2V7-3by  2V7  +  4. 

5.  5Vn-h4by  5Vil-4.  13.  V2  + V3  + V6  by  V2+4V3. 

6.  Vc  +  V2d  by  Vc - V2d.  14.  x - Vxyz  +  2/^  by  Vx  +  V^. 

7.  (-Vac-VS^by,  ^^    V2^-^3^by  ^^-^^. 
a  (3Vi  +  5yi)l  _  ^^       ^^ 
9.  (a  -  a6  V2  +  62)2.  16-  3  V3- V2  by  5  V4  4- V2. 

10.  ^3-3^9  by  ^/3  +  3-^9.    17-  2  V3-^2  by  2  V3- ^4. 

11.  (</9 - 4  ^6 y.  18.  ^ls-2^1  by  V8  +  2V7. 

19.  VlO-3V5-4V3by  4V3-3V5. 

20.  Vic?/  +  2  ViC2!  —  V2/2!  by  ^xy  —  3  ViC2!. 


21.  -\/m^v?-  +  Vm7i  by  -\JmT?  -f-  -^mhi. 

22.  2-^/^-3  vi^s  i^y  a/16^2^^^. 

23.  Vaf"  —  VlO  aj"+^  +  V^  by  V5^  +  V2^«+^. 

24.  V(.'C+?/)*-V(a;+2/)'»+2by  V(a:+2/)""+'+V(a;4-2/)'- 

25.  .+|-^Z^by..  +  |  +  ^/Z±S".        . 


156-157]  IRRATIONAL   NUMBERS— RADICALS  249 

157.  Rationalizing  factors.*  Conjugate  surds.  The  factor 
by  which  ii  given  surd  (radical)  must  be  multiplied  in  order 
to  obtain  a  rational  product  is  called  its  rationalizing  factor. 
Thus,  of  the  surds  V5  a^  and  V  25  a,  each  is  the  rationalizing 
factor  of  the  other,  since  their  product,  5  a,  is  rational ;  the 
same  is  true  of  the  surds  2Va—  V3  and  2 Va  +  VS.     (Why  ?) 

Of  two  such  binomial  quadratic  surds  as  2 Va  —  V3  and 
2Va  +  V3,  which  differ  only  in  the  sign  of  one  term,  each 
is  called  the  conjugate  of  the  other.  Moreover,  since  the 
product  of  any  two  conjugate  quadratic  surds  is  rational 
(§  53),  therefore  each  of  them  is  the  rationalizing  factor 
of  the  other. 

EXERCISE  CIX 

1.  Is  V2a  the  rationalizing  factor  of  V2a?  Why?  Show- 
that  VS  kH  and  2  V3  —  V5  are  the  rationalizing  factors  of  V'^  kH^ 
and  2  V3  +  V5,  respectively. 

2.  Is  V5-2V3  the  rationahzing  factor  of  2V3-hV5? 
Explain.     Are  these  surds  conjugate  to  each  other  ? 

Find  the  rationalizing  factor  of : 

3.  V3.  10.    Vf.  16.    3a-V5^. 

4     ^^7  2  6/"^  17.   ^5x  —  V2ay. 

5.   2ViO.  "'    "'^^'  ^«-    Va»  +  2^6. 


6. 


^■27/    X..  ^.    -  xi-vs- 

'•    ^4*^-  13.    V2-V7.  20.    ^Va-cJl. 

3 _  b  ^  d 

8.  V-i.  14.   4  +  5V3.  ,14-3 

9.  5^^^  15.   2V3  +  V8.         ^^-    \"r'^4^"   • 
22.    How  may  we  find  the  rationalizing  factor  of  any  binomial 

quadratic  surd?     Why?     Does  the  same   method  answer  for  a 
binomial  cubic  surd  ?     Explain. 

*  See  also  §  177. 


250  HIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

158.  Division  of  monomial  radicals.  If  the  dividend  and 
divisor  are  of  the  same  order,  tlieir  quotient  may  be  written 
down  immediately  by  §  149  (ii),  while  if  they  are  of  differ- 
ent orders,  they  should  first  be  reduced  to  equivalent  radicals 
of  the  same  order  (§  154). 

E.g.,  to  divide  -^4  a:(?y^  by  ■\/2  a^x,  we  proceed  thus : 

■V4  aoi?y-  _  V16  a-^m^y^ 


-V2a'x         V8a' 


[§154 


4 


^  CH49(ii) 

W2^^1^^^;^_  [§152 


^ 


159.   Division  of  polynomials  containing  radicals.     If  the 

divisor  is  a  monomial,  then,  manifestly,  the  quotient  may  be 
obtained  by  dividing  each  term  of  the  dividend  by  the 
divisor. 

Ex.  1.   Divide  3V2  +  4V3  by  V2. 

Solution.     ^V2  +  4V3  ^  3  _^  ^^3  ^  3  _^  2V6.  [§  152 

We  may  also  solve  Ex.  1  as  follows : 

3V2  +  4V3^(3V2+4V3)V2^6  +  4V6^o      ^  /^ 
V2  (V2)^  2  -^    ^   ' 

i.e.,  we  may,  before  dividing,  multiply  both  dividend  and  divisor 
by  the  rationalizing  factor  (§  157)  of  the  latter.  This  method 
is  known  as  "  division  by  means  of  rationalizing  the  divisor  " ;  in 
many  examples  it  is  easier  than  the  one  used  in  the  first  solution 
above  (cf.  Exs.  28,  29,  and  37  below). 

EXERCISE  CX 

Eind  the  following  indicated  quotients  and  simplify  each : 

2.  V20--V5.  4.    6V5-T-V4(). 

3.  V2l6-f-Vl2.  5.    V^-f-V3^'. 


1 -is- lot) J  inilATlONAL   NUMBERS  —  RADICALS  251 

6.  V'M-^V^.  13.  6-r-3^4. 

7.  2-v^54--v/216.  14.  VI^TS^f. 

8.  Sv^lS-lS^J.  15.  ^57i2^^-j- V2^. 

9.  </(a-6)---V(a  +  6)l  4a;       3/^ 

10.  2V6-=-^/3.  2/        ^2/' 

11.  2\/l2^V8.  1^-    a^4ar*2/^-^2feA/2a;2/. 

12.  10  --  V5.  18.   3  aV2~W'^^  -^  2  6^3^^. 

19.  Find  the  value  of  1  h-  V2,  correct  to  two  decimal  places : 
(1)  by  finding  V2  and  dividing,  and  (2)  by  first  rationalizing  the 
divisor.     Which  is  the  easier  process  ? 

20.  Find  (correct  to  two  decimal  places)  the  value  of : 

-^  ;   -^  ;  ^  ;  and  -^.     (Cf.  Ex.  19.) 

V3  '    V2  '   V2  '  V(5      ^  ^ 

Find  the  following  indicated  quotients  and  simplify  each : 

21.  ( Vl5  —  V3)  -^  V3.      24.    (x-Vy'z  —  ofyz^)  -7-  ^xyz. 

22.  (V6  +  2V3)--V2.      25.    (-\/aFb--\/'b^  +  -y/d^t)^^abi^. 

23.  (4-7V5)--V6.         26.    (5-v/i2-2V6  +  4)-v--^4. 

27.  In  w^hich  of  the  above  exercises  is  division  by  means  of 
rationalizing  the  divisor  easier  than  direct  division  ? 

Divide  by  means  of  rationalizing  the  divisor,  and  simplify: 

28.  -^ .  31.    t 34.    3V2-4V5. 

V3-V2  Vc-Vd  2V3  +  V7 

29.  ^       .  32.    8:t2V15.        35    g+V^M^-. 
3^^-'^  '    V3+\/5  '    a_V^iM^ 

30.  2  +  V3.  33     27-4V7  3^     V^+^^V^^ 
2-V3                      ■     2V7-1  '     V^+V^^i 

37.  If  the  result  in  Ex.  26  were  wanted  correct  to  3  decimal 
places,  show  in  detail  that  it  is  far  simpler  first  to  rationalize  the 
divisor  than  to  extract  roots  and  divide  by  the  ordinary  arithmeti- 
cal method.     Is  this  true  in  Ex.  28  ?    in  Ex.  33  ? 

HIGH.   SCH.   ALG. — 17 


252  BIGH  SCHOOL  ALGEBRA  [Ch.  XIV 

38.  What  is  the  product  of  (2  +  V3)  -  V5  by  (2  +  V3)  +  V5  ? 
of  this  result  by  2  —  4 V3  ?  What,  then,  is  the  rationalizing 
factor  of  2  +  V3  -  V5?  Divide  2  by  2  +  V3  +  V5;  also  by 
1  4.  V3  -  V2. 

160.  Powers  of  surds.  Roots  of  monomial  surds.  Powers 
of  surds,  being  merely  products  of  two  or  more  equal  surd 
factors,  may  be  found  by  the  method  of  §  155  ;  and  roots  of 
monomial  surds  may  be  found  by  §  149  (iii). 

Ex.  1.    Find  the  fifth  power  of  v'2  a^. 

Solution.     (^27')'  =  V(2^'  [§  151  (iii) 

=  ^2^^=  2  a^^4^  [§  152 

Ex.  2.   Extract  the  square  root  of  -v/4  a^x. 

Solution.     yj'^Ta^x=  i/4r  a^x.  [§  149  (iii) 

This  result  may  also  be  written  in  either  of  the  following  forms  : 

^j■V^o^  or  '\J2  aVx.  [§  149  (iii) 


EXERCISE  CXI 
9. 


Simplify: 

3.  (V'r?)2.  g  (^FA\                 15.   (V2^+V3^)^ 

4.  (2-^aby,  '  ^     ^J                   16.   (5V3-3V2)^. 
5    (</6^m:ii?^'  ^^  (-V2V^)3. 


12 
3,^,,  12.    (^^F^^v^)\  ^^'    (1-^V3). 


7. 


13.    (9+V5)l  19.  (Vm^-Vm7i)2. 

8.    (iViey.  14.    (V7-V2)2.  20.  (V2-3^3)l 

Express  each  of  the  following  by  means  of  a  single  radical 
sign,  and  simplify : 

21.  VV^.  24.    ^-27V^.  27.  VtV-V^^- 

/   o, 3; 7=  28.  -x/VS^ 

22.  V\/3cdl  25.    -^-^V^.  ^ 


___^^  .  29     x\  'fZHH 

23.   V^25mV.        26.   yV(r  +  s)i«^  •    A\8(e-/) 


159-102]  IRRATIONAL    NUMBERS  —  RADICALS  253 

161.  An  important  property  of  quadratic  surds.  Neither 
the  sum  nor  the  difference  of  two  dissimihir  quadratic  surds 
(§  147)  can  be  a  rational  number  ;  for,  if  possible,  let 

Vx±Vy  =  r^  (1) 

-Vx  and  V^  being  dissimilar  surds,  and  r  rational,  and  not 
zero. 

From  Eq.  (1)  ±V^=r-Vi,  (2) 

whence,  squaring,  y  =  r^-~2r  ^x  -h  a;,  (3) 

and,  solving  for  V a:,       ^x  =  — ^ ^ ; 

2  r 

i.e.^  if  Eq.  (1)  were  true,  then  the  surd  ^x  would  equal  the 

rational  number  — ^ ^,  which  is  impossible;  hence  Eq.  (1) 

cannot  be  true. 

From  what  has  just  been  shown  it  at  once  follows  that 
if  x-{-^/y  =  a-\-^^  where  x  and  a  are  rational^  and  V?/  and 
V6  are  quadratic  surds^  then  x  =  a  and  y—h. 

For,  if  re  +  V?/  =  a  +  VJ, 

then  V^  —  V6  =  a  —  x\ 

which,  by  the  above  proof,  can  be  true  only  if  each  member 
is  zero,  i.e.^  ii  a  =  x  and  ■Vy=\^b.  In  other  words,  the 
equation  x  +  Vy  =  a-\-  V3  is  equivalent  to  the  two  equations 
x=  a  and  y  =  h. 

162.  Square  roots  of  binomial  surds.  Some  binomial  quad- 
ratic surds  are  exact  squares;  the  following  examples  show 
how  to  extract  the  square  root  of  such  surds. 

Ex.  1.     Extract  the  square  root  of  8  +  V60. 

Solution.  If  8+V60  is  the  square  of  a  binomial  surd,  let 
Va;+  V2/  represent  that  surd,  i.e.,  let 

then,  squaring,  8  +  V60  =  x-{-2  Vxy -{-y=x-\-y-\-  2^xy ; 

therefore  S  =  x-\-y  and  V60  =  2  ^/xy,  [§  161 


254  HIGH  SCHOOL  ALGEBRA  [Cii.  XIV 

and  combining  these  last  two  equations  (after  squaring  the  second) 
easily  leads  (§  131)  to  the  solution 

x  =  8  and  2/  =  5 ; 

therefore  ^js  +  V60  =  V3  +  V5, 

as  is  easily  verified  by  squaring  the  expression  V3  +  VB. 

Note  .  This  example  might  also  have  been  solved  by  inspection ;  for, 
writing  8  +  V60  in  the  form  8  +  2^V'l5,  and  then  comparing  it  with 
(\/x+  \/?/)2,  i.e.,  with  x  -\-  y  -\- 2  y/xy,  we  see  that  we  have  only  to  find 
two  numbers  whose  sum  is  8  and  w^hose  product  is  15,  and  take  the  sum  of 
their  square  roots  as  the  required  root. 

Ex.  2.   Ky  inspection,  find  the  square  root  of  18  —  Q-y/^. 

Solution.  Writing  18  —  ^y/b  in  the  form  18  —  2  •\/45,  we  see 
that  we  need  to  find  two  numbers  whose  sum  is  18  and  whose 
product  is  45,  and  take  the  difference  of  their  square  roots.  These 
numbers  are  evidently  3  and  15,  hence 

Vl8-2V45  =  V3-Vl5. 

EXERCISE  CXIl 

Find  (by  inspection  where  practicable)  the  square  root  of  each 
of  the  following  expressions,  and  check  your  results : 

3.  4  +  2V3.  a  30-20V2.        13.  e4-4/-4Ve/: 

4.  16  +  2j^l5.  9.  39-12V3.        14.  2 m  +  9 n - 6 V2m?i. 

5.  12  +  8V5.  10.  47-12VI1.      15.  146-56V6. 

6.  17-12V2.  11.  63  +  24V5.        16.  m  +  2Vm, 

7.  27-4V35.  12.  Sxy-4.xyV^.  17.  a  —  Ve. 

18.  In  the  first  solution  of  Ex.  1  above,  why  does  a;  +  2/  =  8, 
and2V^=  VeO? 

19.  If  the  numerical  value  of  '\21-f  8V5  is  required,  is  it 
easier  to  find  first  the  binomial  whose  square  is  21  +  8V5,  or  to 
begin  by  extracting  the  square  root  of  5  ?  Explain.  Also  answer 
the  question  if  12  +  3V5  is  substituted  for  21  -f-  8  V5, 


102-10:1]  IRRATIONAL   NUMBERS  —  RADICALS  255 

163.  Irrational  equations.  Equations  which  contain  indi- 
cated roots  of  the  imkyioivn  numbers  are  called  irrational 
equations  (also  radical  equations) .   Thus  6  ■\/x  —  25  a;  +  88  =  0, 

ViTl+a;  =  8,^^7r-4-l  =  0,  and   3  +  iVi  =  ^a;2-l  are  ir- 
-vx 

rational  equations,  but  such  an  equation  -d^  x—  V3  =  5  a;  is 

rational. 

The  solution  of  irrational  equations  is  illustrated  by  the 
following  examples : 


Ex.  1.   Solve  the  equation  V.^•  +  1  +  .t  =  11. 

Solution.     On  transposing,  the  given  equation  becomes 


■\/x  -h  1  =  11  —  a?, 

whence,  squaring  both  members  (Ax.  3), 

fl;  +  l  =  121-22a;  +  «2, 
i.e.,  a;2_23a;  +  120  =  0, 

whence  (§  126),  a;  =  15  or  8 ; 

and,  on  substitution,  it  is  found  that  15  satisfies  the  given  equa- 
tion if  Vcc  + 1  means  the  negative  value  of  this  root,  while  8  satis- 
fies it  if  the  positive  value  of  this  root  is  intended. 


Ex.  2.    Solve  the  equation  V5  x-\-l  —  wx  +  2  =  3. 
Solution.     On  transposing,  the  given  equation  becomes 
V5  a;  4- 1  =  3  +  Va?  +  2, 
whence,  squaring  both  members  (Ax.  3), 

5  i»  + 1  =  9  +  6  V^T2  +  a;  +  2, 
i.e.,  '         4a;-10  =  6V^T2; 

whence,  dividing  through  by  2,  then  squaring  and  simplifying, 

4a^-29aj  +  7  =  0, 
from  which  (§  126),  a;  =  7  or  ^. 

On  substitution  it  is  found  that  the  given  equation  is  satisfied 
by  x  =  7  if  each  radical  is  regarded  as  positive,  and  by  a;  =  ^  if 
V5  a;  + 1  is  taken  as  positive  and  Va;  +  2  as  negative. 


256  HIGH   SCHOOL  ALGEBRA  [Ch.  XIV 


EXERCISE  CXIII 

Solve  the  following  equations,  and  show  what  restrictions,  if 
any,  must  be  made  on  the  signs  of  the  radicals  in  order  that  your 
results  shall  be  roots : 

16.    -\/3-\-x-\-Vx=—='      . 


3. 

V2a;-f  6  =  4. 

4. 

V4aj  +  5  =  7. 

5. 

V3  a;  -  8  =  Va;. 

6. 

-y/x'-4x  =  2^4:. 

7. 

i/x-\-oc'  =  -2c. 

8. 

9. 
10. 

^5  —  x  =  x—5. 

X  4-  V^  =4:X  —  4  Va;, 

2/4- V^-20  =  0. 

17.  Vm  +  5+ Vm— 8  =  V3. 

18.  Vl+sVs2  +  12  =  l+s. 


19. 


20. 


^v  —  8      Vv  —4 


V -v -}- 1      -Vv—2 


2^     V3a;  +  l  +  V3^^o^ 


11.  V4y  +  17  +  Vy  +  1=4.  ^■^-    V3¥+l-V3^- 

12.  V25-65=8-V25+66.  ^^     V^-2^V^4-1. 

13.  cV^— d\/a?=c-  +  d2— 2cd.  '     ^x-\-3      Vx-\-2 

14.  Va;+1+— 4^  =  2.  23.    r+V^"^^ 

Va;+1  vr  — c^ 

15.  V^^^ ^  =  0.  24.    a-Va^^-a^^_V3-l 

Vs  +  7  a  +  Va^-a^         V3  +  1 


^=-:^ 


25.    V4  a;  +1  —  Vaj  +  3  =  Va;  —  2. 


26.    Va;  +  a  +  Va:  +  ft  =  V2  a;  +  a  +  6. 


27.    Va;  +  3  + V4a.'  +  1  =  Vl0a;  +  4. 


28.    ^^-^^^-^  =  ^+V^^38. 

a;  -  Va;^  -  8 


29-    y'-y-  V2/'-2/  +  4  =  8  (cf.  Ex.  18,  p.  206). 

30.   a  +  10  =  2 Va~+10  +  5. 
31.   How  many  roots  has  the  equation  in  Ex.  27,  if  the  radicals 
are  unrestricted  in  sign?     How  many,  if  each  radical  must  be 
taken  positively  ? 


CHAPTER  XV 
IMAGINARY  NUMBERS* 

164.  Definitions.  In  the  solution  of  quadratic  equations, 
and  elsewhere,  such  numbers  as  V— 5  and  6  —  V—  10  fre- 
quently present  thenjselves;  these  numbers  cannot  be  ex- 
pressed, even  approximately,  as  the  quotient  of  two  integers 
(cf.  §  146).  _ 

Numbers  of  the  form  V—  5,  where  h  represents  a  positive 
number,  are  called  pure  imaginary  numbers,  while  numbers 
of  the  form  a  ±  V  —  6  are  called  complex  imaginary  num- 
bers, or  complex  numbers.  Two  complex  numbers  are  said  to 
be  conjugate  if  they  differ  only  in  the  signs  of  their  imaginary 
terms.  Thus,  V  —  5,  2  V  —  6,  and  V—  |^  are  pure  imaginary 
numbers,  while  2—  V— 3,  7  — 2V— 5,  and  7-f-2V— 5  are 
complex  numbers,  the  last  two  being  conjugates  of  each  other. 

From  the  definition  of  the  symbol  Va  (§  113)  it  follows 
that  (V^=T)2=-5;  (1) 

and  by  the  method  of  proof  used  in  §  150,  it  is  easily  shown 
that  V^=V6.V"^^.  (2) 

Remark.  The  second  member  of  Eq.  (2)  may  be  regarded  as 
a  standard  form;  and  it  will  be  found  that  operations  with  imagi- 
nary numbers  are  usually  much  simplified  by  first  reducing  such 
numbers  to  this  standard  form.  The  symbol  V—  1  is  often 
called  the  imaginary  unit,  and  is  represented  by  /. 


*  Teachers  who  prefer  less  work  in  imaginaries  than  is  here  given  may 
omit  §§  166-168  ;  the  entire  chapter  should  be  read  if  time  permits. 

257 


258  HIGH  SCHOOL  ALGEBRA  [Ch.  XV 

165.  Powers  of  V—  1,  i.e.,  of  i.  As  a  particular  case  of 
Eq.  (1),  §  164,  we  have 

similarly  (V- 1)3=  (V- 1)2.  V-l=-V-l,  "     i^= -i, 

(V^:n)*=(V^^i)3.  V^i:=l,  "    ^4=l, 

(V^a)5  =  (V^^)*- V^^  =  V^,        "     ^5  =  ^, 

and  so  on  for  the  higher  powers  of  V  —  1 ;  any  one  of  these 
powers,  when  simplified,  will  be  found  to  have  one  or  another 
of  the  four  values  :  V  — 1,  —1,  —  V  — 1,  and  1. 

EXERCISE  CXIV 

1.  Define  an  imaginary  number  (of.  §§114  and  146). 

2.  Which  of  the  following  are*  imaginary  numbers :  V—  3, 
V^,  -V^^  V5,  ^^^  3-\/^  4  aV^,  and  I  +  i  V^=^  ? 

3.  Is  V— a;  imaginary  when  x  represents  a  positive  number  ? 
when  X  represents  a  negative  number  ?  Answer  the  same  ques- 
tions for  -^  —  x. 

4.  Eeduce  to  the  standard  form  (§  164,  Kemark) :  V— 9 ; 
V^^;  V"=^l0";   V^^l3;  V^^12;  and  V-i:5c^. 

5.  Show  by  the  method  of  proof  suggested  in  §  150  that 
V^^T^ V7  .  V^l. 

6.  Show  that  if  ^  =  V—  1,  then 

^2=-l,  ^«=-l,  ^l»=? 

^3=  -I,  i'^=-i,  ^"=  ? 

i*=l,  i^  =  l,  i^'=? 

7.  Since  any  even  number  may  be  written  in  the  form  2  n, 
where  n  is  an  integer,  and  since  a^"*  =  (ci^)"?  show  that  every  even 
power  of  i  is  real. 

8.  When  n  is  even,  does  i'-^"  equal  1  or  —  1  ?  Why?  Answer 
the  same  questions  if  7i  is  odd. 

9.  Give    the    values    of    the   following    even    powers  of  i: 

^8[  =  (^2y];     P;    |-32.    ^-18.    ^'64.     ^100.    ,^-6.    ^^^ 


166-167] 


IMAGINARY  NUMBERS 


269 


10.  Show  that  every  odd  power  of  i  is  either  ^  or  —  i  (cf.  Ex.  7). 

11.  Find  by  inspection  the  value  of  the  following  odd  powers 


of  i :  i"^ ;  f 


i'\ 


12.   Distinguish  between  pure  and  complex  imaginary  numbers, 
and  give  three  examples  of  each. 


-B 


A' 


I     I     I. 


B' 


166.  Graphical  representation  of  imaginary  numbers.  Since 
opposite  numbers,  such  as  +  3  and  —  3, 
are  represented  graphically  by  opposite 
distances,  such  as  OA  and  OA' ;  and  since 
multiplying  +3  by  —1  gives  —3;  there- 
fore we  may  regard  the  multiplier  —  1  as 
an  operator  which  rotates  OA  through  180° 
about  0,  into  the  position  OA' , 

Again,  since  i  •  ^=  —1,  therefore  i  may  be  regarded  as  an 
operator  which  when  applied  twice  in  succession  rotates  OA 
through  180°;  hence  using  i  as  a  multiplier  once  (instead  of 
twice)  should  rotate  OA  through  90°  into  the  position  OB. 

In  other  words  :  3v^  — 1  {i.e.^  OA  •  i)  may  be  represented 
graphically  by  OB.  Similarly,  any  pure  imaginary  number 
whatever  may  be  laid  off  on  the  line  B'  OB,  above  the  origin  0 
if  the  number  is  positive.,  below  the  origin  if  it  is  negative. 

E.g.,  if  each  division  on  the  lines  JT  and  RN  in 
the  figure  represents  a  unit,  then  OS=l,  0J=  —  2, 
OL  =  V^  ON  =  SV"^^,  0Q=-  2 V^  etc. 

The  lines  JT  and  BN  are  often  called 
the  axis  of  real  numbers  and  the  axis  of 
imaginaries,  respectively. 


J  H 


o 
p 

Q 

R 


S    T 

-i — H- 


167.  Graphical  representation  of  complex 
numbers.  A  complex  number,  such  as  5  +  3V—  1,  may  be 
graphically  represented  as  follows :  lay  off  OA,  5  units  on 
the  axis  of  real  numbers,  and  OB,  3  units  on  the  axis  of 
imaginaries,  tlien  complete  the  parallelogram  AOBR,  and 
draw  its  diagonal   OR ;  this  diagonal  is  a  graphical  repre- 


260 


HIGH  SCHOOL  ALGEBRA 


[Ch.  XV 


sentation  of  the  complex  number  5  +  3V—  1,  i.e.,  of  the  sum 

of  5  and  3V^=a. 

Moreover,  it  is  evident  that  any  com- 
plex number  vrhatever  may  be  graphic- 
ally represented  by  the  above  method. 


B 

R 

-/^i 

a 

O     1     1     1  J^ 

Note.  The  appropriateness  of  calling  OR  the 
sum  of  OA  and  OB  will  be  evident  to  pupils  who 
have  an  elementary  knowledge  of  physics.  Thus,  if 
two  forces,  represented  in  amount  and  direction  by  OA  and  OB,  respectively, 
act  simultaneously  on  a  body  at  0,  the  result  is  the  same  as  though  a  single 
force  represented  in  amount  and  direction  by  OB  were  acting  on  this  body  ; 
i.e.,  the  sum  of  the  forces  OA  and  OB  is  the  force  OB. 

168.  Graphical  representation  of  the  sum  of  complex  num- 
bers ;  also  of  their  difference.  The  sum  of  two  complex 
R  numbers,  such  as  7  +  2V—  1  and 
1  +  4v^—  1,  may  be  graphically  repre- 
sented as  follows :  let  OP  and  OQ 
be  the  graphical  representations  of 
7+2  V^^nr  and  1  +  4V  -  1,  respec- 
tively (§  167)  ;  complete  the  par- 
allelogram POQR,  and  draw  its 
diagonal  OR ;  then  OB  is  the  graph- 
ical representation  of  the  sum  of  7  +  2V—  1  and  1  +  4V—  1 
(cf.  §167,  Note). 

Obviously  the  sum  of  any  two  complex  numbers  may  be 
represented  by  this  method. 

Again,  to  find  the  difference  of  two  complex  numbers 
graphically,  we  have  only  to  reverse  the  sign  of  the  subtra- 
hend, and  proceed  as  in  addition. 


EXERCISE  CXV 
Represent  graphically  the  following  imaginary  numbers : 

1.  2V^^.  3.    -7 1.  5.  2AL  1.   V^5. 

2.  -5V^.         4.  \i.  6.   V^9.  a   V^. 


167-160] 


IMAGINARY  NUMBERS 


261 


Perform  the  following  additions  and  subtractions  graphically 
9.  4  +  2  [cf.  §  4].  13.  3-7. 

10.  4.i  +  2i.  14.  3^-10^. 

11.  Ai-2i.  15.  3-8-6. 

12.  7 


5.  16.  V-25  4-2V-36-V-49. 

Represent  by  drawing  each  of  the  following  complex  numbers : 

17.  2  +  4aA3i.  21.  4  +  3l  25. 

18.  4  +  2V^^.  22.    -6-6i.  26. 
2 


V-i  +  6. 

V-2-7. 


19 


23.  3+V-36. 

2  +  V^^. 


27.    lO+V^^OS. 
~8. 


5V-1. 

20.    -7  +  1.  24.    -:^  +  V-9.  28.    -i^- 

Perform  the  following  indicated  operations  graphically : 

29.  (l+4i)  +  (4  +  2i).  33.   (5-V^~9)  +  (-3  +  V^^). 

30.  (3  +  2i)  +  (5  +  3i).  34.    -2^  +  (-3  +  V^^). 

31.  (2-50  +  (8  +  0.  35.   (4  +  3t)-(2  +  i). 

32.  (3  +  V^4)  +  (-2  +  V^).   36.   (8-20-(5-3i). 


37.  (10+V-9)-(-2+V-16). 

38.  (|_V38)_(7-V^. 

169.  Fundamental  operations  with  pure  and  complex  imagi- 
nary numbers.  If  complex  numbers  are  first  reduced  to  the 
standard  form  a  +  5V— 1,  and  if  we  are  careful  to  remember 
that  V  —  1  •  V  —  1  =  —  1  and  not  +  1,  then  the  operations  of 
addition,  subtraction,  etc.,  with  these  numbers  may  be  per- 
formed exactly  as  are  the  corresponding  operations  with 
real  numbers.  The  following  examples  will  illustrate  these 
operations. 


Ex.  1.  Find  the  sum  of  2  +  V-  9,  8  -V^i,  and  -  V -  25. 

Solution.     Since    2  +  V^9  =  2  +  3  V^^, 

8-V=l  =  8-2V"^, 

and  -V^r25=    -5V^^, 

10_4V^  i.e.,  10-V^=36. 


therefore  the  sum  is 


262  Jiiaii  SCHOOL  algebra  [Ch.  xv 

Ex.  2.     Multiply  r>V^^  by  W^^. 

Solution.  5V^  =  5 V2  •  V^, 

and  4  V^^  =  4 V7  .  V^=l, 


hence  the  product  is  20  V2  •  V7   (V  — 1)^  i.e.,  — 20Vl4. 

Note.  Observe  that  this  product  is  not  +20Vl4,  as  it  would  be  if  tlie 
factors  were  i-eal  numbers.  Beginners  should  be  especially  careful  to  guard 
against  errors  in  the  sign  of  a  product  of  imaginary  numbers. 

Ex.  3.    Multiply  3  +  V^^  by  2  -  V^^. 

Solution.  Writing  these  imaginary  numbers  in  terms  of  the 
imaginary  unit,  the  work  may  be  arranged  thus : 

3H-V5.  V^ 

2-V3.  yiTi 

6  +  2V5.V^^ 

-  3  V3  ■  V^  -  Vl5  (V3i)2 
6+ (2  V5-3  V3)  .  V=3  + Vi5. 


Ex.4.   Divide  12  + V- 25  by  3- V- 4. 

Solution.  Such  divisions  are  easily  performed  by  first  mul- 
tiplying both  dividend  and  divisor  by  the  conjugate  of  the 
divisor,  thus : 


12  +  V-  25  ^  12  +  5  V  ^=n:  ^  (12  +  5  V^I)(3  +  2  V^:i) 
S-V"^^        3-2V^=^       (3-2V^(3  +  2V^ri) 
^36  +  39  V^i:  +  10  ( V"^)' 
9  _  4  ( V^^)' 
26  +  39V^^ 


9  +  4 


2  +  3V-1. 


170.  Important  property  of  complex  numbers.  By  a  method 
altogether  like  that  used  in  §  161  it  may  be  shown  that  if 
a  +  5V— 1  =  ^  +  (^V—  1,  then  a  =  c  and  b  =  d.  Moreover, 
this  fact  may  be  used,  as  in  §  162,  to  extract  the  square  roet 
of  any  complex  number  (cf.  Ul.  Alg,  §§  151,  182). 


16l>-170j  IMAGINARY  NUMBERS  263 

EXERCISE  CXVI 

5.  Add  3  +  5  i  and  7-|- v  — 4  as  in  §  169;   then  add  these 
numbers  graphically,  and  check  your  work  by  comparing  results. 

Simplify  Exs.  6-13  below,  and  check  as  teacher  directs : 

6.  7-6^+2  +  3^.  8.    (3  +  2  i)  -  (3  -  2  *). 

7.  (3  +  2*)  +  (3-2i).         9.    (-4-A/^^=^)+(-4-V^. 


10.    V-4  +  4V-9  +  V-25. 


11.   3  +  V-16  +  V-4-5-V-9. 


12.   3  +  V-36-(l+2  V-25)  +  3  V-16. 


13.    V-49  +  5  V-4-(6  +  2  V-9). 
Simplify  each  of  the  following  expressions  (cf.  §  162): 
14.    V^--(2V^  +  o~3  V^24)  +  3V^^18. 


15.    V-16  a^x"  +VI-5  +  2V0-3O-V-9  a2«2  +  V-  aV. 


16.  a;  V— 4+V  — ar'  — 2if  — 1  — V— 32. 

17.  Solve  the  equation  x^  —  1  =  0  (cf.  §  72)  and  find  the  sum  of 
the  roots;  check  your  addition  graphically.  Similarly  find  the 
sum  of  the  3  cube  roots  of  8;   of  the  3   cube  roots   of   —27. 

Find  the  product  of : 

18.  3  V^=^  by  5  V-12.  20.    2  V^^  by  V-4aV. 

19.  5  V^^  by  2  V^^.  21.    —1-6  r^  +  i*^  by  i\ 

22.  V^^  +  V^=^  by  V"^  -  V^=^ 

23.  3+2V^^by  5-4V^=n:. 

24.  V^r50_2V^^n^by  V^^-5V^^. 

25.  3P-4.i^hy2i^-3i^\ 

26.  Show  that  the  sum,  and  also  the  product,  of  a-\-bi  and 
a  — pi  is  real  (a  and  h  being  any  real  numbers).  Show  that  the 
same  is  true  also  for  V—  4  —  3  and  —  V  —  4  —  3. 

27.  Show  that  both  the  sum  and  also  the  product  of  any  two 
conjugate  complex  numbers  is  real. 

28.  Multiply  V^a  +  V^^  +  V"^  by  V^^  —  V^^  +  V^^ 


264  HIGH   SCHOOL  ALGEBRA  [Ch.  XV 

29..  (l-}-V^)'=?  30.  (2-3if=?     31.  (2a-3a;V^)'=? 

32.  Find  the  product  of  a  V— ^  +  6  V— ot,  a  V— a  +  6  V— 6, 
and  h  V—  h  —  a^  —  a. 

33.  Show  that  —  ^  +  i  V—  3  and  —  i  —  i  V  —  3  are  conjugates 
of  each  other,  and  also  that  each  is  the  square  of  the  other. 

34.  Reduce  -^ ~    -\ —     ^    to  its  simplest  form. 

3-V^4       3  +  2i 

35.  Simplify  each  of  the  following  indicated  quotients : 

V^T5.  V^24.  V^.   Vc     V^.    V84     v:r6-i-2V^:r8~ 
V^^^'  V"^^^'  V^=^'  V^^'   V5  '  2V^^'        v^^ 


36.    Show  that  ^  +  ^^zl  =  «c  +  ?>^  +  (^c-ad)V-l . 

c+dV-1  ^^^ 

Perform  the  following  indicated  divisions  (cf.  Ex.  4),  and  check 
your  work  by  multiplying  the  quotient  by  the  divisor : 

37.     -i-,.  40.    ^-±^^. 

l^i  5-2z 

38     — 41     V2a;-3ai 

^■'  +  ^'  '    V2^  +  26^ 

39.    2W^.  ^2     V^-lV6. 

3  +  V-2  ?:V6H-Va 

43.  If  a  and  h  are  positive  and  unequal  numbers,  show  that 
V—  a  ±  V—  6  cannot  equal  a  real  number  (cf.  §  161). 

44.  Show  that  if  cc  +  V— 2/  =  aH-V— 6,  wherein  y  and  h  are 
positive  numbers,  then  x  =  a  and  ?/  =  ft. 

45.  Find  the  square  root  of  5  —12  V—  1. 

Hint.    Let  y/x-y/y.  V^^l  =  Vs  -  12  V^HT  (cf.  §  162). 

Find  the  square  root  of : 

46.  10-6V^^.  48.    3  +  2V^=l0. 

47.  6V^^-17.  49.   5|-3}V^^. 


CHAPTER   XVI 

THEORY   OF    EXPONENTS 

ZERO,   NEGATIVE,   AND  FRACTIONAL  EXPONENTS 

171.  Introductory,  (i)  As  originally  defined  (§9)  an 
exponent  is  necessarily  a  positive  integer,  and  it  is  in  this 
sense  only  that  we  have  thus  far  used  it.  Under  this  restric- 
tion we  have  established  the  following  exponent  laws  (§  110), 
wherein  a  is  any  real  number  except  0  : 

I  dT"  'dP'  =  a"^+», 

II  {^ory  =  a*»% 

III  {ahy  =  d^  .  5% 

IV  a"'  :  a»  =  a^-**. 

(ii)  We  now  propose  to  extend  the  meaning  of  an  expo- 
nent so  as  to  include  such  symbols  as  a^  a~^,  and  a^,  along 
with  our  former  exponent  expressions. 

In  extending  the  meaning  of  any  symbol  already  in  use, 
however,  the  extended  meaning  should  be  such  as  not  to 
disturb  any  rules  of  operation  already  established  for  the 
symbol  in  question.  Hence,  we  shall  admit  such  symbols  as 
a^  a"^  etc.,  into  our  algebraic  notation  if,  and  only  if,  we  can 
assign  to  each  of  these  symbols  a  meaning  consistent  with  the 
above  exponent  laws. 

172.  Meaning  of  such  symbols  as  a^,  a~^  and  ai     (1)   If, 

following  the  plan  given  §  171  (ii),  wa  let  ?^  =  0  in  law  I, 
§  171,  we  obtain 

a"*   ^0  =  a"*,  [since  a^^^  =  a^ 

hence  a^  =  1  \ 

i.e.,  if  law  I  is  to  admit  the  symbol  a^  then  this  symbol  must 
have  the  value  1. 

266 


266  HIGH  SCHOOL  ALGEBRA  [Ch.  XVI 


(2)  Again,  if  in  law  I,  §  171,  we  let  m  =  5  and  n=—3, 
we  obtain  a^  -  a~^  =  a^  =  1,  [since  a^~^  =  a^ 

whence  a~^  =  — ; 

aP 

i.e.,  if  law  I  is  to  admit  the  symbol  a~^,  then  this  symbol 
must  have  the  same  meaning  as  — . 

(3)  And  finally,  if  law  I,  §  171,  is  to  remain  valid  for  such 
symbols  as  a^,  then 

a^  -a^  -a'  =  a\  [since  a^^'^'^^  =  or 

and  therefore  a^  =  Va\  [§11^ 

i.e.,  if  law  I  is  to  admit  the  symbol  a%  then  this  symbol  must 
have  the  same  meaning  as  ~va^. 

p 
173    Definitions  of  a^,  a"^  and  a^.     Using  as  a  basis  the 

special  cases  considered  in  §  172,  we  shall  now  define  the 

p 
symbols  a^  a~\  and  a""  ?ls  follows: 

(1)  aO=l, 

(2)  «-'  =  -,. 

a'' 

and  (3)  a^-=</'^; 

wherein  a  represents  any  number  whatever,  and  k,  p,  and  r 

are  positive  integers. 

Moreover,  these  definitions  —  since  they  are  based  upon 
one  exponent  law  only  —  must  be  regarded  as  tentative  until 
they  are  shown  (§§  174,  175)  to  satisf}^  all  of  the  exponent 
laws. 

EXERCISE  CXVII 

Assuming  the  validity  of  §  173  (1)  and  (2),  show  that : 

1.   3c-^=3.i=^^.  3.    a%'c=c. 

c      c 


172-174]  TIIEOliY  OF  EXPOXIJXTS  267 

Keduce  the  following  to  equivalent  expressions  free  from  zero 
and  negative  exponents : 

5.  ««.  9.    5A-2.  13.   Sx'^-r-y-*. 

6.  cc-3.  10.   ifc-^.  14.   f-^y-\ 

7.  Qa\  11.    7--r-3.  15.   2/'-2/"^ 

8.  -4-!-a°.  12.    Sa^V'-  16.   3  a^-^^ -- ic^/-^. 

Assuming  the  validity  of  §  173  (3),  translate  the  following  into 
equivalent  radical  expressions : 

17.  ai  20.    t^.  23.  (ia-^^s)!. 

18.  ai                             21.    {axy^.  ^*'  (^^"'>^- 
19     a;l                               22.    (4c^d)i  25.  (^^'y. 
Write  in  the  fractional  exponent  notation  : 

26.  a/^.  30.    -v/oV.  34.    -V-^m^. 

27.  -v/m.  31.    %V.  35^    &-^'l6^. 

28.  V 6.  36.    2\x    y-^. 

^\  23  _i 

29.  V^24.  33.    \-^,'  37.    acv^a^c-^^. 

38.  Find  the  numerical  value  of:  5« ;  3"^;  4'^;  9*  ;  (i)^  ;  4-2-2^. 

39.  Show  that  if  law  II,  §  171,  is  to  admit  the  symbol  a^,  then 
a^  must  equal  1  (cf.  §  172). 

40.  Show  that  if  law  IV,  §  171,  is  to  admit  the  symbol  a"'*, 

then  a~^  must  equal  —• 
a 

41.  Show  that  if  law  II,  §  171,  is  to  hold  for  a^,  then  a^  must 
equal  Va^. 

174.*    The  symbols  a^  and  a~*  obey  all  the  exponent  laws. 

That  a^  and  a^^  as  defined  in  §  173,  satisfy  all  the  exponent 
laws  may  be  shown  by  assigning  zero  and  negative  integral 

*The  proofs  given  in  §§  174  and  175  may,  if  the  teacher  prefers,  be  omitted 
until  the  subject  is  reviewed. 

HIGH  8CH.  ALG.  —  18 


268  HIGH  SCHOOL    ALGEBRA  [Ch.  XVI 

values  to  m  and  n^  both  separately  and  together,  in  the  equa- 
tions which  express  those  laws. 

Thus,  if  we  let  w  =  0  in  law  II  (n  remaining  a  positive 
integer),  we  obtain  (a^y  =  a^  [since  a"  • «  =  a" 

i.e.,  r  =  l, 

which  is  correct  ;  hence  a^  =1  is  consistent  with  law  II. 

Again,  let  m  =  0  in  law  I V  (ri  remaining  a  positive  integer), 
and  we  obtain  a^ :  a""  =  a~\  [since  aO-"=  a~" 

i.e.,  l:a^  =  a-% 

which  is  a  correct  equation   [§  173  (2)]  ;  hence  a^  =  l  and 

a""  =  —  are  consistent  with  law  IV. 

Once  more,  let  m  =  —  r  and  n=  —s  (where  r  and  s  are 
positive  integers)  in  law  II,  and  we  obtain 

(a-0"*  =  «""""' =  «'■*; 
but  this  is  consistent  with  §  173  (2),  for 


<-=(r=^ 


[§  173  (2) 
[§§  109,  92 


or 

Moreover,  if  we  similarly  test  the  remaining  combinations 
of  positive  and  negative  integral  and  zero  values  of  m  and 
71,  in  the  four  exponent  laws,  we  find  that  definitions  (1)  and 
(2)  of  §  173,  and  the  exponent  laws  of  §  171,  are  entirely 
consistent.  Hence  we  need  no  longer  regard  the  definitions 
of  a^  and  a~*  as  tentative  (cf.  §  173,  last  part). 

The  testing  of  some  or  all  of  these  remaining  cases  may 
be  assigned  as  an  exercise  to  the  pupil. 

p. 
175.  The  symbol  a'  satisfies  all  the  exponent  laws.     That 

p        

a^=:^aP  is  consistent  with  the  exponent  laws  may  be  shown 

as  follows  : 


174-175]  THEORY   OF  EXPONENTS  269 

Let  p,  r,  s,  and  t   represent  any  positive   integers,  then 

PI         .  _ 


=  Va^^  •  Va''^=  VaP'  •  a''         [§§  154,  149  (i) 

pt+rs  P,i 


^.e.,  law  I  holds  good  for  such  symbols  as  a^, 

p   s^  s  

Similarly,  (</')'  =  (  V^)' = ^(  Va^)^ 

=  V^  [§  149  (iii) 

ps  Pi. 

P 
i.e.,  law  II  holds  good  for  such  symbols  as  a^, 

p        .        

Again,  (a6)^=  </(^ahy=<JaP  •  6^ 

=  7^.V^  [§149(i) 

p  p 

2 

i.e.,  law  III  holds  good  for  such  symbols  as  a^. 

The  proof  for  law  IV,  being  closely  similar  to  that  for  law 
I,  is  left  as  an  exercise  for  the  pupil. 

Observe  that  the  above  proofs  remain  valid: 

(1)  if  r  (or  t)  takes  the  value  1,  in  which  case  ^  [or  -  j  be- 
comes an  integer. 

(2)  if  p  (or  8)  is  negative  or  zero  (cf.   §  174),  in  which 

^(  or  -  )  becomes  a  negative  fraction  or  integer,  or  zero. 
r\     tj 

Therefore   these   proofs,  taken   in  connection  with  those 

previously  given,  include  all  possible  combinations  of  positive 

and  negative,  integral,  fractional,  and  zero  values  of  m  and  n, 

in  the  exponent  laws  of  §  171.     We  need,  therefore,  no  longer 

p 
regard  the  definition  of  a'^  [§  173  (3)]  as  tentative. 

p 
Note.    Observe  also  that  a!"  represents  the  principal  rth  root  of  a^,  since 

that  is  the  meaning  of  the  symbol  y/(f  (§  150). 


case 


270  HIGH  SCHOOL  ALGEBRA  [Cii.  XVI 

EXERCISE  CXVIII 

1.  Does  a^  equal  x^  even  when  a  and  x  are  unequal  ?    Explain. 

2.  By  means  of  §  173  (1)  show  that  a"*  -r-  a"  =  a"*'"  when  n  =  0 
(cf.  §  171,  IV). 

3.  By  means  of  §  173  (2)  show  that  (a^)-^  =  a-^  and  that 
a-'^a-'  =  a-'  (cf.  §  171,  II  and  IV). 

4.  Show  that  6a!^^  =  6aY  ^^^^  ^^^^^  3:Var^ ^ 2_:^^3 ^ 

y~*  XT  2*xy~^        S^n~^x 

5.  By  proceeding  as  in  Ex.  4  show  that,  by  changing  the  sign 
of  the  exponent,  di  factor  may  be  transferred  from  the  numerator 
to  the  denominator  of  a  fraction,  and  vice  versa. 

6.  Is  ^^^-^ equal  to  ^L±A?    Explain.     Observe  carefully 

4  a?  4:Wx 

that  a  factor  but  not  a  part  may  be  transferred  as  in  Ex.  5. 

Free  the  following  expressions  from  negative  exponents,  and 
explain  your  work  in  each  case : 

7.  ^. 
8. 


5-2 

11. 

52 .  12-1 
10-1 .  3* 

15. 

(82_1)-1 

2d' 

12. 

a-' +  2. 

16. 

9-V 
6x-' 

a-'x-'     , 

13. 

x-'-\-y-^ 
5 

17. 

9-V 

h-'x-' 

(-6)-i(a^2/)-^ 

3.2-2 

8-2 

14. 

m~3  —  n"^ 
m-3 .  w-« 

18. 

3r-2-4s 

10. 

19.  Is  .  ^  equal  to  3  aa;-^  ?     Why  ? 

a;2 

20.  As  in  Ex.  19,  write  the  following  fractions  in  integral  form  : 
o^.      4:a-^c-\      4-ia-V.     m-i-3n-* 

6^2/'  ac     '      5a-id~i'  m~^n 

21.  What  is  the  diiference  in  meaning  between  mJ  and  m^? 
between  m-^  and  m"'^  z.e.,  between  m^  and  m  ^  ? 


175-176]  THEORY  OF  EXPONENTS  271 

Write  Exs.  22-26  below  as  radical  expressions,  and  write  Exs. 
27-30  with  positive  fractional  exponents : 

22.  a~T^6TV.         ^^    6~«/?.  28.    -Vr-Ps^ 

23.  2^dh~'^.  1        n 
26.   a'^  —  b^. 


29.    V-32r-V«. 


24.       — -     .  __       3/ — .   ._«,,_o,        _.       * 


3^'  27.    -v/-ic-«d-'e.      30.    V(c+d)«-f--v^(c+d)*. 

Find  the  numerical  value  of : 
31.   A  35.   6° -2-*.  /27xY 

32.4-  36.    (.09)1.  'V.OOSr 

40.    (32-^^(32*)-^. 

33.  9t.  37.    (256)..  41.   (-,W-(169)i 

34.  8-^.25^        38.    (Hl)"^  42.   (64)-^ .  (16-^)^ 

-(i)'-(r-(f )■'-"■»-■ 

176.  Operations  with  negative,  zero,  and  fractional  expo- 
nents. From  the  definitions  of  negative,  zero,  and  fractional 
exponents  (§§  173-175)  it  follows  that  in  all  operations 
with  such  symbols  as  a^  a~^  and  «s  their  exponents  obey 
the  exponent  laws  of  §  171  ;  that  is,  these  exponents  behave 
just  as  though  they  were  positive  integers. 

When  working  with  fractional-exponent  expressions  it  is 
frequently  necessary  to  change  the  exponents  to  higher  or 
lower   terms  ;    this  does  not  change  the  value  of  such  an 
expression ; 
for  since  V^=  %^,  [§  151  (iv) 

therefore  a'' =  a^'^ . 

Operations  with  radicals  may  often  be  greatly  simplified 
by  first  converting  the  various  radical  expressions  into  their 
fractional-exponent  equivalents  (cf.  Exs.  29,  41,  and  57  in 
the  following  exercise). 


272       -  HIGH  SCHOOL  ALGEBRA  [Ch.  XVl 


EXERCISE  CXIX 

.5 


1.  Simplify :    V2  •  ^4  •  2^  •  2-^ 

Solution.     \/2  .  ^  •  2'  •  2"^  =  2^  •  2^  .  2^  .  2"^  [§  173  (3) 

=  2MH-|=2'3  =  ^4.  [§171,  I 

2.  Simplify:   {ah)^  -  (p^)^. 

Solution.     (a&)^  •  {h'^c)^  =  (ab^c)^  =  ahc^.  [§  171,  III,  II 

6^r 


3.   Simplify:    (- 

s„.„„o..    (_)    =(_)    =(-)    =_  f5„i,„ 

~  &2  -    52    • 

Perform  the  following  indicated  operations,  and  express  your 
results  in  their  simplest  form : 

4.  8t .  8^ .  si       7.  (^)f .  A  y  .  M\     10.  s-^i^-^s=^ 

8.   2x^  -h4:  x^' 
6.  2L8-i.27i      9.^?.^?.  12.  (36a-tci)-i. 

13.  By  means  of  fractional  exponents,  reduce  Va^  and  V^  to 
equivalent  radicals  of  the  same  order  (cf.  §  154). 

2.  5 

Solution.     The  given  radicals  are,  respectively,  equivalent  to  g^  and  cc^' 

and  these  expressions   are,   respectively,  equivalent  to  g^  and  o;^,  i.e.,  to 
\/g4  and  v^,  each  of  which  is  of  order  6. 

14.  Solve  Exs.  12-16,  p.  245,  by  means  of  fractional  exponents. 

15.  By  means  of  fractional  exponents,  solve  Exs.  19-22,  p.  247, 
Exs.  10-14,  p.  251,  and  Exs.  9-12,  25-29,  p.  252. 

By  means  of  fractional  exponents,  simplify : 

16.  V^-^Sv"^  .  -y/x.  18.    ("v/r^  .  Vi  .  -v/O"" 

17.  ("v/m^  .  VOi  19.   (8^  .  8-^)-2. 


176] 

THEOR 

20. 

21. 

-^4a-2-^-8a-^. 

22. 

(^?)-^.(3s-V^-)l 

23.  ViC"^  •  Vit*"^. 

24.  (Vo'^VO"'""- 
i 


THEORY  OF  EXPONENTS  273 


26  (a""  +  o^^)V^. 

27  m^  —  m~^  +  a/7  m^ 

28. !;! 

o 


25.   f    ""  '\      ]  •  V  ^  + 

0,2 


29.  Eind  the  product  of  3Va  —  BVy  by  2Va  +  V^/. 


Solution.     Since  SVa  -  6\/y  =  S  a^^  -^vK  and  2  Va  +  S^y  =  2a^+ ?/3, 
therefore  this  product  becomes 

2a^  +  y'^ 
qJ-^^  -  10  Jy^ 

+  3  Jy^  -  5  y^'^^ 

Qa-7  a^y^  -6y\ 
If  it  is  desired,  this  product  may,  of  course,  be  written  in  either  of  the 
following  forms  :6a-  lVa\/y  —  5\/«/2,  or  6  a  -  iVcfiy^^  —  5v^. 

Perform  the  following  multiplications : 

30.  a^  +  6^  by  a^  —  &i  32.  m^  —  m^r^  -f-  n^  by  m^  +  ?i^. 

31.  oj^  —  Q^y^  +  2/^  ^y  i»^  +  2/^-   33.  m^ — m^n~^ + n"  ^  by  m"^ + n  ^. 

34.  ia^^-TV^^/HTVa^^^Z-aV^/^byiaj^  +  i^/^. 

35.  81  yy^  -  27-s/'^</y  +  9^/^Vf  -  3  2/a/^4-  2/Hy  3  "V^  +  y\ 

36.  V«  —  4-v/a^a;  +  6^/ax  —  4 A^aa;^  H-  Vx  by  \/a  —  2 Vax  +  V«. 

37.  '>n  ^  +  m"^  —  2  m^  +  4  m~^  by  1  +  2  m^  —  -^—  • 

Vm 

3  *_  4  — i 

38.  ^r^  +  q-^-^  —  p-'^^q  ^  by  p~-^^  +  Q"  ^. 

39.  If  Jx^/x  +  2 n Vn  +  f  a.'^«  +  6 n^/^  by  V7i  -  3 a;^  +  ^,«^- 

40.  5  a-'x*  +  3  a%"x-'  -  b'-V  by  x-^  -  3  ^"~%-'  +  ah 


274  HIGH  SCHOOL  ALGEBRA  [Ch.  XVI 

41.   Divide  x^  —  y^hj  Va?  +  Vy. 

Solution.     Since  \/x  +  v^  =  a;^  +  y^,  this  solution  may  be  put  into  the 
following  form : 

x2  —  ?/3 


x3  +  y^ 


x^  —  x^y^  +  xy  -  x'^y^  +  x^y^  -  y^ 


—  x^y^  —  y^ 

—  x^y^  —  x^y 


x^y  -  y^ 
x%  +  xy^ 


-  xy^  -  y^ 

3  2. 

—  xy^  —  x^y"^ 


1 

X^y^  —  y3 

a;  V  +  x^y^ 
-  x^y^  ■ 
-x^y^ 


r 


0 
The  above  quotient  may  also  be  written  thus  : 

VxP  —  Vx^  Vy  -\-  xy  —  y/o^Vy^  +  Vx  •  y^  —  y/y^. 

Note.  To  appreciate  one  of  the  advantages  of  fractional  exponents  the 
student  has  only  to  perform  the  division  in  Ex.  41,  using  the  radical  notation, 
and  compare  his  work  with  the  above  solution. 

Perform  the  following  divisions  : 

42.    a  +  x^hy  a  •  +  x^.  43.   m^  —  n^  by  m^  —  n®. 

44.  x-^  +  3y~^  — 10  xy-'^  by  x''^  Vy  —  2. 

45.  J  +2  ^ah-'^  +  ^  by  ^a  +  6"^. 

b 

46.  x^  +  x^  -\/y  —  X  -y/x  y'^  —  xy-{-  -Vx  y^  +  y'^  by  Va;  4-  -y/y. 
Simplify  the  following  expressions : 

47.  /Va?  +  -v^.vY.  y/x-Vy      ^^        x-y  y^-yt 

a;— +r'*'aj-"-r"**  '  2/  +  V2/+I  '  yt-1 


176J 


51. 


THEORY  OF  EXPONENTS 
1        .        1 


275 


a^ 


+ 


Write  down,  by  inspection,  the  square  root  of  each  of  the 
following  expressions: 

52.  1  — 2i6^  +  wi  54.   p^  — 4  +  4p"i 

53.  x^  +  4  x^  +  4.  55.   ax^  +  2  a^x^  -\-  a^x, 

56.  771 -\- n -{- p  —  2  m^n^  +  2  n^p^  —  2  m^pK 

57.  Extract  the  square  root  of  ^x*  —  2  Va^ '+  5  Va^  —4  Va?  +  4. 
Solution.     This  expression  written  in  the  equivalent  fractional-exponent 


form  is  cc^  —  2  a;'^  4-  5  x^  —  4  x^  +  4,  and  in  this  form  its  square  root  may  be 
extracted  just  as  though  it  were  a  rational  expression  (cf.  §  117)  ;  thus  : 


2  x^  +  5  x5  -ix^  +  A\x^  -xM-  2 


x^  —  x^ 


-2x^ 


+  5x^ 


2x^  +  x 


2x^ 


2x^  +  2 


4x^+4 


4x^ 

4  x^  —  4  x^  4-  4 


0 


hence  the  required  root  is  xs  —  x?  +  2,  i.e.,  Vx^  _  vx  +  2. 

Extract  the  square  root  of  each  of  the  following  expressions : 

58.  ar^  +  2  a;^  +  3  a;  4-  4  a;^  +  3  +  2  a."2-  4-  x'K 

59.  a3  —  4  ot^  +  4  a  4-  2  a^  —  4  a^  +  aK 

60.  n'^  —  2  wr^ii^'  +  2  m^n^  +  imT^n  5—2  mW 4- m^ 

Extract  the  cube  root  of  the  following  expressions ;  write  the 
results  with  all  exponents  positive,  and  then  replace  all  fractional- 
exponent  forms  by  radical  signs  (cf.  footnote,  p.  185): 

61.  8  4-12a;t4-6a;^4-aj2. 

62.  8a;-i-12a;"%4-6aj-V-2/^. 

63.  r^_6ri4-15r^-204-15«^-6^4-^i 

64.  8  a^h--^  4-  9  a&^  4- 13  a^  4-  3  a^6  4- 18  a^h'^^  4-  ^'  4- 12  ah-\ 


276  HIGH  SCHOOL  ALGEBRA  [Ch.  XVI 

Solve  the  following  equations: 

65.  m^  =  4.  67.    x~^  =  5.  69.   a;"^  =  — 27. 

66.  ^^  =  8.  68.    \yi  =  25.  70.    yjm^  =  3\js. 

71.  2r^  +  5r^  — 3  =  0.     [Hint.     Put?/  =  ri]. 

72.  £c"^  4- 5  a;"^  4- 4  =  0. 

73.  (2A;-3)-2  +  7(2A:-3)-i-8  =  0. 

177.  Rationalizing  factors  of  binomial  surds.  Another 
advantage  of  the  fractional-exponent  notation  is  that  it  fur- 
nishes an  easy  method  for  finding  the  rationalizing  factor  of 
any  binomial  surd  whatever,  —  only  quadratic  binomial  surds 
were  considered  in  §  157. 

Ex.  1.   Eind  the  rationalizing  factor  of  x^  +  2/^- 

Solution.  Since  (x'^y  —  (y^y  is  exactly  divisible  by  x'^  -j-  y^ 
when  n  is  any  positive  even  integer  [§  66  (ii)],  and  since  6  is  the 
smallest  value  of  n  for  which  both  (x^y  and  (y'^y  are  rational, 
therefore  the  rationalizing  factor  (cf.  §  157)  is 

(X^y—(yh'^  X^—f  5  4      1  2     3  1,  6 

-^^ — 7 1 —  =  ^ 7  =  a?3  —  x^y^  -\-  xy  —  x^y^  -^  x^y^  —  y^ ' 

.^3 -f  ?/^  x^-\-y^ 

Ex.  2.   Find  the  rationalizing  factor  of  x^  +  y'^. 

Solution.     As  in  Ex.  1  [cf.  %66  (iv)],  we  find  this  factor  to  be 

(X^y^  +  (yh^^  X^  -f  V^^  M  13      2     ,        J  2     4  11 

x'5  _|_  2/^  a;^  -f  y^ 


EXERCISE  CXX 

Find  the  rationalizing  factors  of  the  following  expressions : 

3.   s^+A                         7.    Sv^  —  2vK 

11.    ah'^-^3v\ 

4.    s^  —  t^.                         8.    s~^-\-t\ 

12.    x~^-\-2y\ 

5.    a^  —  xK                     9.    2m^  — ?ii 

13.   .'i;"*  — Zi 

6.   m^  +  ni                  10.   2x^-3yk 

14.   2r~h-'^-t' 

CHAPTER   XVII 

QUADRATIC  EQUATIONS 
[Supplementary  to  Chapter  XII] 

178.  Solution  of  quadratic  equations  by  means  of  a  formula. 

Since  every  quadratic  equation  in  one  unknown  number 
may  be  reduced  to  the  form  ax^ -\-hx-{-  c=  ^  (§  122),  and 

since  the  roots  of  tliis  equation  are ^whatever 

the  number 8  represented  hy  a,  6,  and  c  (Ex.  3,  p.  195),  therefore 
the  roots  of  any  particular  quadratic  equation  may  be  found 
by  merely  substituting  for  «,  5,  and  <?,  in  the  roots  of  the 
above  general  equation,  those  values  which  these  coefficients 
have  in  the  particular  equation  under  consideration. 


E.g.,  since  the  roots  of  ax^ +  hx+  c  =  {)  are    — ^  "^       —,  therefore 

the  roots  of  3  ic2  +  10  x  -  8  =  0  (in  which  a  =  3,  &  =  10,  and  c  =  -  8)  are 


-10±VlO-^-4.3.(-8)      ,,,.^-10^14     ,.,.,2,^^_4, 
2.3  '        '  6         '  3 

So,  too,  the  roots  of  6  y^  4. 19  ^  _  7  =  0  are 


19±Vl9^-4.6(-7)  1  ^^^  _  7 

2.6  '        ' 3  2 


and  the  roots  of  a;2  -  3  x  +  5  =  0  are  ,-(-3)=tr  V(- 3)2 — 4.1-5^ 

i& "  1 

i.e.,  ^ 

Note.  While  the  student  should,  of  course,  be  able  to  solve  quadratic 
equations  without  the  use  of  the  formula,  he  is  advised  to  commit  this  for- 
mula to  memory,  and  henceforth  to  employ  it  freely  ;  he  will  find  this  well 
worth  his  while,  because  roots  of  quadratic  equations  are  so  often  required 
in  mathematical  investigations. 

277 


278  HIGH  SCHOOL  ALGEBRA  [Ch.  XVII 

179.   Character  of  the  roots  of  ax"  -\-bx  +  c  =  0.   Discriminant. 

As  we  have  already  seen  (§  126),  the  roots  of  the  equation 

^     ,  ^  -b-h  -VP-^ac        ,    _5_  V62-4a(? 

aar  +  oa;  +  <?  =  0  are  ^ and  t: 

2a  2a 

Hence  it  follows  that,  if  a,  5,  and  c  represent  real  and 
rational  numbers,  the  roots  can  be  imaginary  or  irrational 
only  if  V52  — 4ac  is  imaginary  or  irrational;   ^.e., 

If  h^  —  4tac  is  positive^  the  roots  are  real  and  unequal; 
ifb^~4:ac  =  0,  the  roots  are  real  and  equal ; 
if  l^  —  ^  ac  is  negative^  both  roots  are  imaginary/  ; 
and  the  roots  are  rational  only  when  b^—iac is  an  exact  square, 
(Let  the  pupil  fully  explain  each  case.) 

The  expression  b'^  —  4:ac^  which  determines  the  character  of 
the  roots,  is  usually  called  the  discriminant  of  the  equation. 

Thus,  without  actually  solving  the  equation  3  cc^  _  5  aj  —  l  =  0,  we  know 
that  its  roots  are  real,  irrational,  and  unequal  because,  for  this  equation, 
\/&2  _  4  ac  =  V37,  and  VST  is  real  and  irrational. 

Similarly,  we  see  that  the  roots  of  2 r.^  +  5 a;  —  8  =  4a;  —  11  are  imaginary, 
because  for  this  equation  h"^  —  Aac  =  —  23. 

EXERCISE  CXXI 

1.  By  means  of  the  formula  of  §  178,  solve  Exs.  6-17,  p.  196. 

Without  first  solving  the  following  equations,  tell  whether 
their  roots  are  real,  imaginary,  rational,  equal,  etc.,  and  explain : 

2.  a?-^x  +  Q>  =  0.  5.   3^^4-ll^  +  17  =  0. 

3.  a;2-6a;  +  9  =  0.  6.   1(3.^2+ 2) -i  =  i (a; -5). 

4.  ^^-1^-11=0.  7.   25?^2_20^^  +  7  =  0. 

8.  In  each  of  Exs.  4-11,  p.  197,  determine  the  character  of  the 
roots  without  solving  the  equation. 

9.  For  what  value  of  h  will  the  roots  of  3  .t^  — 10  ic  +  2  A;  =  0 
be  equal  ? 

Suggestion.     The  roots  will  be  equal  if  (-  10)2  _  4  •  3  .  2  A;  =  0.     Why  ? 


179-180]  QUADBATIC  EQUATIONS  279 

For  what  values  of  m  will  Exs.  10-15  have  equal  roots  ? 

10.  mic^— 6x4-3=0.         13.  my^  —  5my-\-ll  =  m. 

11.  ^  +  3ma;  +  7  =  0.        14.  y\l  - m)  -\-7 y  =  9  -3my. 

12.  3a^-4mic-f  2  =  0.      15.   -4i/2-32/-3  =  m(2/-f  2/  +  4). 

16.  Translate  into  verbal  language  the  conditions  for  the  char- 
acter of  the  roots  of  a^i?  -\-hx-\-G  —  0. 

17.  Show  that  if  one  root  of  a  quadratic  equation  is  imaginary, 
then  both  are  imaginary,  and  each  is  the  conjugate  of  the  other. 

18.  For  what  values  of  A;  are  the  roots  of  36  a?^  —  24  A;a;  -f- 15  A; 
=  —  4  imaginary  ? 

Solution.  Here  the  discriminant  Ifi  —  ^ac=  (  —  24  ky-  —  4  •  36(15  Jc  -\-  i) 
=  144(4  A;2  -  15  A;  -  4)  =  144(4  A;  +  1)  (k  -  4)  [cf.  §  64].  Hence  the  roots  are 
imaginary  for  those  values  of  k  which  make  {4:k  +  l)(k  —  4)  negative,  and 
for  those  values  only.  Now  (4A;  +  l)(^•  —  4)  is  negative  only  when  one  of 
its  factors  is  positive  and  the  other  negativ^;  hence  the  roots  of  the  given 
equation  are  imaginary  when  k  lies  between  —  ^  and  4.     (Why  ?) 

19.  For  what  values  of  k  are  the  roots  of  36  f  —  24:kt-\-15k 
=  —  4  real  ?     How  do  the  roots  compare  if  k=  —  ^?    k  =  4:? 

20.  Without  actually  solving  the  equation,  find  the  values  of 
m  for  which  the  roots  of  4  mV  +  12  m^x  -\-10  =  m  are  equal ;  also 
those  values  of  m  for  which  the  roots  are  real ;  also,  those  for 
which  the  roots  are  imaginary. 

180.   Relation  between  roots  and  coefficients.     If  we  let  r 

and  /  represent  the  roots  of  ax^  -^  bx  -\-  c  —  0^  i.e,^  if 


_54.V52-4a(?       ■.    ,      -b-Vh^-iac 

2  a                                        2  a 

then    r  +  r'  = 

_5  +  V^2_4^^      _5_V52_4ac_      6, 
2a                             2a                     a 

and        r-r'^ 

_  J  4-  V52  _  4  ac      -b-  V62  -4:ac_c 

2a                             2a                 a 

(2) 

Let  the  pupil  work  out  (1)  and  (2)  in  detail,  and  translate 
each  of  these  equations  into  verbal  language. 


280  HIGH  SCHOOL  ALGEBRA  [Ch.  XVII 


EXERCISE  CXXII 

In  the  following  equations,  name  the  sum  of  the  roots,  also 
their  product ;  then  check  your  answers  by  solving  the  equations. 

1.  x'  +  5x-2  =  0.  4.   a;2_30a;  +  25  =  0. 

2.  a^-10x  =  —  16.  5.    a^-\-px  =  -q. 

3.  4s2-6s=3.  6.   ax^-{-2bx^c  =  0. 

7.  In  each  of  Exs.  4-11,  p.  197,  write  down,  without  solving  the 
equation,  the  sum  and  the  product  of  its  roots;  explain  your 
work  in  each  case. 

8.  If  one  root  of  ic^  +  5  a;  —  24  =  0  is  known  to  be  3,  how  may 
the  other  root  be  found  from  the  absolute  term  ?  from  the  co- 
efficient of  the  first  power  of  a;  ?     Do  the  results  agree  ? 

9.  If  one  root  of  any  given  quadratic  equation  whatever  is 
known,  how  may  the  other  root  be  most  easily  found  ? 

10.  What  is  the  sum  of  the  roots  of  3  mV  -j-(Sm  —  l)x  4-5  =  0? 
For  what  value  of  m  is  this  sum  3? 

11.  If  one  root  of2a^  —  3(2A:  +  l)a;-}-9A;  =  0  is  the  reciprocal 
of  the  other,  find  the  value  of  k. 

Hint.    Equate  one  root  to  the  reciprocal  of  the  other  and  solve  for  k. 

12.  For  what  value  of  k  will  one  root  of  the  equation  in 
Ex.  11  be  zero  ?  With  this  value  of  Zc,  what  will  be  the  value  of 
the  other  root? 

13.  Answer  the  questions  of  Exs.  11  and  12  for  the  equation 

2(k+iyx'-S(2k  +  l)(k  +  l)x-{-9k=0. 

14.  Show  that  if  one  root  of  ax^  -\-bx-\-c  =  0  (whatever  the 
values  of  a,  b,  and  c)  is  double  the  other,  then  2  6^  =  9  ac. 

181.   Values  of   simple   expressions   containing   the    roots. 

If   r  and   /   are  the  roots  of   a  given    quadratic  equation, 
§  180  enables  us  to  find  the  value  of   such  expressions  as 

-  +  — ,  r^  +  r'^,  etc.,  without  first  solving  the  equation. 


180-182]  QUADRATIC  EQUATIONS  281 

E.g.,  the  value  of  -  +  — ,  for  the  equation  fl:;^  —  5  a;  -f  3  =  0,  may 
be  found  thus : 

1      l_r|-f-r. 

r     r        rr 
but,  for  this  equation,      r'  +  r  =  5  and  ri^'  =  3, 

therefore  -  +  -  =  -. 

r     r'     3 

Similarly,  since  r^  +  r'^  =  (r-{-  r'y  —  2  rr'j 

therefore,  for  the  above  equation, 

7^4-/2  =  25-6  =  19. 

182.   Formation  of  equations  whose  roots  are  given  numbers. 

(i)  Sum  and  product  method.     If  r  and  r'  are  the  roots  of 

the  equation  ax^  -\-  bx  +  e  =  0,  i.e.,  of  a^-\--x-\--=0,  then 

a         a 

(§  180)  this  equation  may  be  written  in  the  form : 
2^  —  (r  +  r'}x  -\-  rr'  =  0. 
And  from  this  we  learn  how  to  write  down  a  quadratic 
equation  whose  roots  are  any  two  given  numbers. 
E.g.,  if  the  roots  are  to  be  2  and  —  5,  we  have 
—  (r  +  r')  =  3,  and  rr'  =  —  10, 
whence  the  equation  is 

a^4_3aj-10  =  0. 

(ii)   The  factor  method.     An  equation  whose  roots  are  any 
given  numbers  may  be  written  down  as  follows  (cf .  §  72)  : 

The  roots  of  (^x  — r}(x  —  r')  =  0 

are  evidently  r  and  /  ;  hence  the  equation  whose  roots  are 
2  and  -  5  is  (x  -  2)(x  +  5)  =  0, 

i.e. ,  as  before,  a^  -^Sx  —  lO  =  0. 

EXERCISE  CXXIII 

1.   If  r  and  r'  denote  the  roots  of  x^  —  lx  +  12  =  0,  find,  with- 
out solving  the  equation,  the  value  of 

J-  J-  -I-       ,        -L  9       ■  19 

——,,  —,,  -  +  -,  r^-\-r". 

ly.  _l_  y.1  n.iy.1  n.  y>^ 


282  HIGH  SCHOOL  ALGEBliA  [Cii.  XVII 

Find  for  each  of  tlie  following  equations  the  value  of  -  +  -  , 
11  r     r" 

T^  +  ?''^,  r  —  r',  and '- 

r     r 

2.  f~-12t  =  -m.  5.   a;2  +  2a;  =  4. 

3.  6m2  — m  =  2.  6.    si?  -{-px-\-q  =  0. 

4.  32/^-162/  +  5  =  0.  7.   ax^  +  hx-\-c==0. 

8.    Solve  each  of  the  above   equations  and   thus  verify  the 
results  in  Exs.  2-7. 

By  each  ol  the  methods  given  in  §  182,  form  the  equations 
whose  roots  are : 

21.    1±V5. 

16.  r^,  r'^.  2 

17.  V6±4.  22.   5,  -^. 

18.  5±2V3.  23.    Vc±V^. 

14.   r,  —  r'.  19.   2  ±  5  ?:.  24.    r±-,- 

r 

25.  The  roots  of  x'  —  ^x-\-2  —  0  being  ?♦  and  ?•',  form  a  new 

equation  whose  roots  are  -  and  -.     (Cf.  Exs.  1  and  15  above.) 

r  r' 

26.  If  r  and  r'  are  the  roots  of  3z^  —  llz-20  =  0,  form  an 

equation  whose  roots  are  -  and  _  ;  also  one  whose   roots   are  r^ 

r  r' 

and  ?''^;  also  one  whose  roots  are  r  -\ — andr'  +  -. 

r'  r 

Write  the  equations  whose  roots  are : 

27.  -  1,  2,  -  5.  29.    1  ±  V5,  5. 

28.  -a,  -b,  —c.  30.    ± Vc,  c  +  c?. 

183.   Factors  of  quadratic  expressions.     As  in  §  182,  if  r 
and  r'  are  the  roots  of  the  equation  ax^  -{-bx-{-c  =  0,  then 

a^  +  -  X  +  -  =  x^  —  (r  -{-  r')  X  -\-  rr'  =  (x  —  r^Cx  —  rO, 
a         a  V  yv  ^ 


9. 

5,  -3. 

10. 

-4,4. 

11. 

-a,  -6. 

12. 

i|. 

13. 

i-f 

182-184]  qUADUATlC  EQUATIONS  288 

^.e.,  multiplying  by  a^ 

ao^  -\-  hx  -\-  c  =  a  (x  —  r)(x  —  r^)  ', 
hence,  if  the  roots  of  the  equation  aa^  +  bx-i-  c  =  0  are  r  and 
r\  then  the  factors  of  the  expression  ax^  -{- bx -\-  c  are  «,  x  —  r^ 
and  x  —  r'. 

184.     A  quadratic   equation  has   two  roots,  and  only  two. 

By  actually  solving  the  equation  ax^  -\-  bx  +  c  =  0  (§  126) 
we  find  that  it  has  two  roots,  say  r  and  r^  That  it  can 
have  no  other  root,  as  r",  is  evident  if  we  write  the  equation 
in  the  form 

a(x-r')(x-r^)=0  [§  183 

and  observe  that  a  (r"  —  r}(r"  —  r'}  cannot  be  zero  if  r'' 
differs  from  both  r  and  r', 

EXERCISE  CXXIV 

1.  Since  2  and  7  are  the  roots  of  x^  —  9  x-\- 14:  =  0,  what  are 
the  factors  of  a^  —  9  ic  + 14  ? 

2.  By  first  finding  the  roots  of  the  equation  15  a^  —  4a?— 3  =  0, 
find  all  the  factors  of  the  expression  15  a^  —  4  a;  —  3.  Check  your 
answer  by  finding  the  product  of  these  factors. 

3.  Write  a  carefully  worded  rule  for  factoring  quadratic 
expressions  by  the  method  of  §  183. 

Find  (in  accordance  with  the  rule  just  made)  all  the  factors  of 
the  following  expressions,  and  check  your  results : 


4. 

5a;2-12a;-9. 

11. 

(2y-iy-5(y  +  l)+8. 

5. 

Sz'-4:5z-lS. 

12. 

a^+px-^q. 

6. 

a)2  +  9. 

13. 

ax^  +  bx-\-c. 

7. 

s^+s+l. 

14. 

3m2              ^ 
-— m  —  5. 

8. 
9. 

4m2-24m-13. 
8a^_2a;-3. 

15. 

10. 

(^  +  l)(2-a^)+9 

HIGH    8CH.    ALG.— 

—  X. 
•19 

16. 

m^  +  6  m  + 13. 

284  HIGH  SCHOOL  ALGEBRA  [Ch.  XVII 

17.  Are  the  expressions  in  Exs.  4-16  equal  to  0?     What  justi- 
fication have  we,  then,  for  writing  them  so  ? 

18.  How  many  roots  has  a  quadratic  equation  ?     Verify  your 
answer  for  the  equations  28  x^  -f-  29  a;  +6  =  0  and  m^— 10  m=  —25. 

19.  Show  that  the  cubic  equation  27  2/^  —  1  =  0  has  three  roots 
and  only  three  (cf.  Ex.  17,  p.  263). 


REVIEW  EXERCISE- CHAPTERS  XJ-XVIi 

1.  Find  the  square  root  of  x^ -\- 20  a^ -\- 16  —  4:  oc^ -{- 16  x ;  also 
of  a;'*  —  2  x^y~^  —  4  cc^  +  y~^  +  4  x^y  +  4  xy^. 

2.  Find  (correct  to  three  decimal  places)  the  value  of  Vll-7 ; 
VA;  V23561. 

3.  Expand  by  the  binomial  theorem  :  (a^  —  2/^" ;   (  9"  ~^  ^ 
(l  +  c-y-  (l-Sm^y-,  ^-l 


+  a;i 

4.  Use  the  binomial  theorem  to  find  (correct  to  five  decimal 
places)  the  value  of  (10.001)^  i.e.,  of  (10  +  .001)^ 

Simplify : 

5.  ^1  +  -^^+^.  7.    </ab^(ah-'c^)(a-h^c-'). 


prnj 


\  z-l  ^  z  +  1  Xc**     V'c^     V 

9.    (a3-62)^(-^^-V6). 

^c-d^c-d     ^c-{-d^c  +  d 


11.    (2V-3-3V^=^)(6V-2  +  4V-3). 


13.   By  rationalizing  the  divisor  perform  the  following  indi- 
cated divisions : 

1  +  r  +  Vl^^.  1  V^-V^^3y  .  3 

l+r-Vl-r^'     V6-V3-I'     V^^-V^'     3*-f5"^ 


184] 


q  UADRA  ric  Eq  ua  tiojsi  s 


285 


14.  Express  -^^89  —  28 VlO  as  the  difference  of  two  surds. 

15.  Find  the  sum  of  V- 25,  - 1- V^  5-lOV^,  and 
—  2  —  7  i,  graphically ;  also,  by  the  same  method,  subtract  the 
fourth  of  these  numbers  from  the  third. 


Solve  the  following  equations  and  check  as  the  teacher  directs 

20.   m  +  5  -h  Vm  -\-5  =  6. 


16.    f-|  +  7|  = 


17.     ^x-^l  +  x  = 


18.    i-?l 


=  1. 


19     3  a^  4-  V4  a?  —  a?^  ^  2 


21.  ^a-{-z-{--\/a  —  z  =  -y/b. 

22.  aj^  +  4  ic^  —  117  =  0. 

23.  19a;*4-216a;^  =  a;. 


24.   144r2-l  +  6V9r2-r=16r. 


3  aj  —  V4  a;  —  x^ 

25.    (7-4  V3)a:2+(2-V3)a;-2  =  0. 
Solve  the  following  systems 
0^2  +  2/2  =  85, 
,  xy  =  42. 


26. 


29. 


(2 


xy  =  4:-  y\ 
0^-2/2  =  17. 


27. 


r  0^-2^^  =  215, 
I  a^4-a^?/  +  2/'  =  43. 


28.     ^ 


f  3r2-2rs  =  15, 


30. 


31. 


a?      2/ 

,a^    r 

mo  4-  Vv  -h  ty  =  11, 


t2r  +  3s  =  12. 


2vw—  Vv  +  w  =  13. 


32.  Show  that  the  difference  of  the  roots  of  the  equation 
ic^  +pa;  +  g  =  0  is  the  same  as  that  of  a52  -|- ^px  +  2 p^  +  g  =  0. 

33.  For  what  values  of  m  will  a^  -  2  (1 +3  m)  «+  7  (3  +  2  7^)  =  0 
have  real  roots  ?    equal  roots  ?    imaginary  roots  ? 

34.  Form  the  equation  whose  roots  are  a,  —  a,  and  h ;  also  the 
equation  whose  roots  are  the  negative  reciprocals  of  the  roots  of 
ax^  —  6a;  4-  c  =  0. 

35.  If  the  roots  of  a^  —  2>a5  +  g  =  0  are  two  consecutive  integers, 
show  that  2>'  —  4  g  —  1  =  0. 


286  HIGH  SCHOOL   ALGEBRA  [Ch.  XVII 

36.  A  rectangular  plot  of  ground  contains  42  acres;  find  its 
sides  if  its  diagonal  measures  1243  yards. 

37.  In  a  regiment  drawn  up  in  the  form  of  a  solid  square, 
the  number  of  men  in  the  outside  five  rows  is  -fj  of  the  entire 
regiment.     Find  the  size  of  the  regiment. 

38.  In  a  quarter  of  a  mile  drive,  the  fore  wheel  of  a  carriage 
makes  22  revolutions  more  than  the  hind  wheel ;  if  the  circum- 
ference of  each  wheel  were  2  ft.  less  than  it  now  is,  the  fore  wheel 
would  make  33  revolutions  more  than  the  hind  wheel.  Find  the 
circumference  of  each. 

39.  A  crew  can  row  a  certain  course  upstream  in  8^  minutes, 
and  were  there  no  current,  they  could  row  it  in  7  minutes  less 
than  the  time  it  now  takes  them  to  drift  downstream.  How  long 
would  it  take  them  to  row  the  course  downstream  ? 

40.  Two  men,  A  and  B,  have  a  money  box  containing  $  210, 
from  which  each  draws  a  certain  fixed  sum  daily,  the  two  sums 
being  different.  Find  the  sum  drawn  daily  by  each,  knowing 
that  A  alone  would  empty  the  box  5  weeks  earlier  than  B  alone, 
while  the  two  together  empty  it  in  6  weeks. 


CHAPTER   XVIII 
INEQUALITIES 

185.  Definitions.     The  symbols  >  and   <   stand  for  "is 

greater  than,"  and  "  is  less  than,"  respectively ;  thus,  the 
expression  a<b  is  read  ''a  is  less  than  5."  One  real  number, 
a,  is  said  to  be  greater  than  another,  6,  U  a  —  b  is  positive ; 
if  a  —  5  is  negative  then  a  is  less  than  b. 

E.  g.,  5  <  8,  since  5  -  8  =  -  3  ;  and  -  4  >-  9,  since  -4-(—  9)  =  +5. 

A  statement  that  one  of  two  numbers  is  greater  or  less 
than  the  other  is  called  an  inequality ;  thus,  5a;— 3>2yis 
an  inequality,  of  which  5  a;  —  3  is  the  first  member,  and  2  y 
the  second. 

Two  inequalities  are  said  to  be  of  the  same  species  (or  to 
subsist  in  the  same  sense')  if  the  first  member  is  the  greater  in 
each,  or  if  the  first  member  is  the  less  in  each ;  otherwise 
they  are  of  opposite  species. 

Thus,  the  inequalities  a  >  6  and  c  +  <?  >  e  are  of  the  same  species,  while 
2-2  ^  y'2  ->  2;2  and  m^<,n^  +  mn  are  of  opposite  species. 

186.  General  principles  in  inequalities.  Before  memoriz- 
ing the  following  principles  (1-7),  the  pupil  should  illus- 
trate each  by  one  or  more  numerical  examples;  he  should 
also  try  to  invent  a  proof  of  his  own  for  each  principle  before 
reading  the  printed  proof. 

Principle  1.  If  the  same  number  is  added  to,  or  sub- 
tracted from,  each  member  of  an  inequality,  the  result  is  an 
inequality  of  the  same  species. 

287 


288  HIGH  SCHOOL  ALGEBRA  [Ch.  XVIII 

For,  if  a<h,  i.e.^  if  a  —  h  is  negative,  then  a-{-  c  —(b  -\-  c^^ 
which  equals  a  —  b^  is  negative,  and  therefore 

a  -f  c<b-^  c, 
and  similarly  a—  c<b  —  c. 

So,  too,  if  « > 5,  then  a-\-  c>b  -\-  c^  and  a—  c>b  —  c. 

Principle  2.  If  each  member  of  an  inequality  is  multi- 
plied or  divided  by  the  same  positive  number^  the  result  is  an 
inequality  of  the  same  species. 

For,  if  a>b^  and  n  is  any  positive  number,  then  an  —  bn, 
i.e.^  (a  —  b^n^  is  positive  (why?),  and  therefore  an>bn. 

Similarly  if  we  divide  by  n ;  and  so,  too,  \i  a<b. 

Principle  3.     If  each  member  of  an  inequality  is  multi 
plied  or  divided  by  the  same  negative  number^  the  result  is  an 
inequality  of  opposite  species. 

Hint.  If  «  >  6,  and  n  is  negative,  then  an  —  hn  is  negative  (why  ?)  and 
therefore  an  <  hn. 

Principle  4.  If  several  inequalities  of  the  same  species 
are  added^  member  to  member^  the  result  is  an  inequality  of  the 
same  species. 

Hint.  Let  a<b,  c<d,  e</,  ...,  then  (a  -  b)-\-(c  -  d)  +  (e  -/)+  •••, 
i.e.,  (a  -f  c  +  e  +  •••)  — (6  +  d  +/+  •••))  ^^  negative  (why  ?),  and  therefore 

Principle  5.  If  an  inequality  is  subtracted  from  an  equa- 
tion^ or  from  an  inequality  of  opposite  species.^  member  from 
member.,  the  result  is  an  inequality  whose  species  is  opposite  to 
that  of  the  subtrahend. 

Hint.  Let  a<Cb,  and  c  =  d  or  c^d,  then  in  either  case,  b  —  a  -\-  c  —  d, 
which  equals  c  —  a  —(d  —  b),is  positive  (why  ?),  and  therefore  c  —  a^d  —  b. 

Principle  6.  If  the  first  of  three  numbers  is  greater  than 
the  second.,  and  the  second  is  greater  than  the  third.,  then  the 
first  is  greater  than  the  third  ;  and  conversely. 

Hint.  Let  a  >  6  and  6  >  c,  then  (a  —  &)  +  (&  —  c),  i.e. ,  a  —  c,  is  positive 
(why  ?),  and  therefore  a>c. 


186-187]  INEQUALITIES  289 

Pkinciple  7.  If  two  or  more  inequalities  of  the  same 
species^  whose  members  are  positive^  are  multiplied  together^ 
member  by  member^  the  result  is  an  inequality  of  the  same  species. 

Hint.  Let  a  >  6  and  c>  d,  then,  by  Prin.  2,  ac  >  he  and  he  >  hd^  whence, 
by  Prin.  6,  aC^^hd  -,  and  similarly  for  three  or  more  such  inequalities. 

187.  Conditional  and  unconditional  inequalities.  An  iden- 
tical or  unconditional  inequality  is  one  which  is  true  for  all 
values  of  its  letters.  Thus,  a  +  4  >  a  and  (x  —  yY  +  1  >  0 
are  unconditional  inequalities. 

A  conditional  inequality  is  one  which  is  true  only  on  con- 
dition that  certain  restricted  values  are  assigned  to  its  letters. 
Thus,  a;-f4<3a;— 2  only  on  condition  that  x>'^. 

A  conditional  inequality  is  solved  by  means  of  the  princi- 
ples of  §  186,  and  in  much  the  same  way  that  an  equation  is 
solved  by  means  of  the  ordinary  axioms. 

Ex.  1.     If  3 .'»  —  -2^  >  -^  —  X,  find  the  possible  values  of  x. 

Solution.  On  multiplying  each  member  of  this  inequality  by 
3,  it  becomes  9»=-25>ll -3x,  [§186,2 

whence  9  a;  +  3  a;  >  11  +  25,  [§  186,  1 

I.e.,  12  a;  >  36, 

whence  ic> 3 ;  [§  186,  2 

i.e.,  if  the  given  inequality  is  true,  x  must  be  greater  than  3. 

By  means  of  the  principles  established  in  §  186  the  student  may  show  that 
each  step  in  the  above  reasoning  is  reversible,  and  hence  that  the  converse 
is  also  true  ;  viz. ,  that  if  a;  >  3,  then  3  x  —  ^^  >  ^^^  —  ic. 

rx-         ^i_    .  1  .•         (2a;  +  32/>5,  (1) 

Ex.  2.     Given  the  two  relations  \  .         /  /o\ 

I     a;  +  42/  =  6;  (2) 

to  find  those  values  of  x  and  y  that  will  satisfy  them  both. 

Solution.  On  multiplying  each  member  of  (1)  by  4,  and  each 
member  of  (2)  by  3,  we  obtain 

8a;-}-122/>20, 
and  3a;-|-122/  =  18; 

whence,  subtracting,  5  a?  >  2,  [§  186,  1 

and  therefore  a^  >  f •  [§  186,  2 


290  HIGH  SCHOOL  ALGEBRA  [Ch.  XVIII 

Now  substitute  for  x  in  (2)  above,  any  number  greater  than  |, 
and  find  the  corresponding  value  of  y  (this  value  of  y  will 
always  be  less  than  J ;  why  ?)  ;  these  values  of  x  and  y,  taken 
together,  will  satisfy  both  (1)  and  (2). 

EXERCISE  CXXV 

3.  Using  the  definitions  of  "greater"  and  "less"  in  §  185, 
show  that  5  >  2 ;  that  -  23  <  -  12 ;  and  that  2  >  -  9. 

4.  It  x-^y  >  z  —  iv,  show  that  x-\-w  ^  z  —y. 
Hint.    Apply  Principle  1  twice. 

5.  May  terms  be  transposed  from  one  member  of  an  inequality 
to  the  other  ?     If  so,  how  and  why  (cf .  Ex.  4)  ? 

^    j^    m-n^m-\-2n^    ^^^^    ^^^^    3(m -ii)  <  2(m  +  2  n). 

How  may  an  inequality  be  cleared  of  fractions  ?     Why  ? 

7.  Show,  from  Principles  2  and  3,  how  to  remove  a  common 
factor  from  both  members  of  an  inequality. 

8.  What  happens  if  the  signs  in  each  member  of  an  inequality 
are  reversed  ?     AVhy  ? 

Hint.    In  the  proof  of  Principle  3,  put  —  1  for  n. 

9.  If  a  >  6  and  c  —  d,  show  that  G  —  a<d  —  h. 

10.  Illustrate  by  numerical  examples  that 

(1)  if  a  >  &  and  c  <  d,  then  the  sum  of  these  inequalities  may 
be  either  a  +  c  =  64-d,  ora  +  c>&4-(?,  or  a  +  c<54-d; 

(2)  if  a  >  6  and  c  >  d,  then  the  difference  of  these  inequalities 
may  be  either  a— c  =  h^d,  a  —  c^h  —  d,  ova  —  c<h—d. 

11.  Translate  (1)  and  (2)  of  Ex.  10  into  verbal  language. 

12.  If  a  <  6  and  c  <  d,  and  if  d  alone  is  positive,  show  that 
ac  >  hd.     Is  this  inconsistent  with  Principle  7  ? 

13.  If  a,  b,  c,  and  d,  are  positive  numbers,  and  if  a  >  6  while 
c<d,  which  is  the  greater,  ac  or  bd  ?  Why  ?  Illustrate  your 
answer  numerically. 

14.  What  operations  Avith  or  upon  inequalities  lead  to  results, 
one  of  which  is  certainly  greater  or  less  than  the  other  ? 


187]  INEQUALITIES  291 

15.  Name  and  illustrate  some  operations  with  inequalities 
which  lead  to  results  whose  relations  are  uncertain. 

16.  Show  that  a^  +  b->2  ah  except  when  a  —  h. 
Hint,     (a  —  6)2  is  positive  whether  a>  6  or  a  <  6. 

17.  Distinguish  between  a  conditional  and  an  unconditional 
inequality.  To  which  of  these  classes  does  a^  +  h^  -\-l>2  ah 
belong  ?     Why  ? 

18.  To  which  class  of  inequalities  does  6  £c  —  5  >  3  a?  belong  ? 
AVhy  ?     Solve  this  inequality. 

19.  If  3  a;  <  5  X  —  9,  show  that  x  is  greater  than  4^  (cf.  Ex.  1). 

20.  If  x^  -\-2^  <C  11  Xj  show  that  the  range  of  values  of  x  is  be- 
tween 3  and  8.  i.e.,  that  x  must  be  greater  than  3  and  less  than  8. 

Hint.  In  order  that  (x  -  3)  (8  —  x),  i.e.,  \\x-  x'^  -  24,  may  be  positive, 
both  factors  must  be  positive  or  both  negative. 

Find  the  range  of  values  of  x  in  each  of  Exs.  21-26 : 

21.  x"  <  9. 

22.  a.-2-h24>lla;.  25. 

Q^  10  — a;>5. 

23.  30>x-f4^>25.  ) 

2  r3-4a^<7, 

24.  28>3a;  +  x2.  26.    |^_^2<4. 

27.  Show  that  no  positive  number  plus  its  reciprocal  is  less 

than  2  5  i.e.,  n  being  any  positive  number,  that  n  4-  -  <  2.* 

n 

28.  Show  that  4  ic-  +  9  <  12  ic. 

29.  Show  that  2  6  (6  a  -  5  6)  >  (2  a  +  &)(2  a-h). 
If  a,  &,  and  c  are  positive  and  unequal,  show  that 

30.  a'  4-  ?>'  >  a%\  32.  a^  +  6^  >  a^ft  +  ah\ 

g-     a4-26      a4-3  ?>         33.  a" -{■  h" -\- c^ > ah -\- he  +  ca. 

'    a-\-Sb      a  +  46         34.  a^-{-]/-^cy^>3ahc(Gt'Ex.30,]).51). 
35.    If  a=^  -f  &2  =  1^  and  (r'  +  cJ-  ==  1,  prove  that  ah-\-Gd>l. 

*  Cf.  Ex.  16.  The  symbol  <  stands  for  "  is  not  less  than,"  and  >  stands 
for  "  is  not  greater  than," 


4x-ll>| 


292  HIGH  SCHOOL  ALGEBRA  [Ch.  XVIII 

36.  If  both  m  and  *i  are  positive,  which  is  the  greater, 
m-i-n  ^^   2mn  ^ 

2  m  4-  ?i 

Solve  the  following  systems : 

37      (2x-Sy<2,  (y-x>9, 

\2x-\-5y=6.  39.    |7^_^j_^^ 

r3  a;  +  2  2/  =  42,  I  20      15 

I  7  1 4  a;  <  3  2/. 

41.  Find  the  smallest  integer  fulfilling  the  condition  that  ^  of 
it  decreased  by  7  is  greater  than  J  of  it  increased  by  6. 

42.  The  sum  of  three  times  A's  money  and  4  times  B's  is  $  1 
more  than  6  times  A's ;  and  if  A  gives  $  5  to  B,  then  B  will  have 
more  than  6  times  as  much  as  A  will  have  left.  Find  the  range 
of  values  of  A's  money  and  B's. 


CHAPTER   XIX 

RATIO,    PROPORTION,  AND  VARIATION 

I.    RATIO 

188.  Definitions.  The  ratio  (direct  ratio)  of  two  numbers 
is  the  'quotient  obtained  by  dividing  the  first  of  these  num- 
bers by  the  second.  The  numbers  themselves  are  usually 
called  the  terms  of  the  ratio,  the  first  being  the  antecedent, 
and  the  second  the  consequent. 

E.g.^  the  ratio  of  15  to  5  is  15  -^  5,  i.e.,  3  ;  the  antecedent  is  15,  and  the 
consequent  is  5. 

The  ratio  of  a  to  5  may  be  vsrritten  as  «  :  6,  a  -^  5,  or  - ; 
it  is  read  "the  ratio  of  a  to  6,"  or  "a  divided  by  6." 

The  inverse  ratio  of  6k  to  5  is  6  -h  a,  i.e.^  it  is  the  reciprocal 
of  the  direct  ratio  of  these  numbers. 

Two  numbers  are  said  to  be  commensurable  or  incommen- 
surable with  each  other  according  as  their  ratio  is  rational  or 
irrational  (cf.  §  146),  i.e.^  according  as  they  have  or  have  not 
a  common  measure. 

E.g.^  1.5  and  f  are  commensurable  with  each  other;  so  also  are  3\/2  and 
5  v'2  ;  but  3V^  and  5  are  incommensurable. 

189.  Ratio  of  like  quantities.  The  concrete  quantities 
with  which  algebra  is  concerned  are  expressed  by  means  of 
numbers,  and  the  ratio  of  two  like  *  quantities  is  therefore  the 
ratio  of  the  numbers  which  represent  these  quantities. 

E.g.,  the  ratio  $6  :  $9  is  the  same  as  6  :  9,  i.e..,  as  2  :  3. 

*  Unlike  quantities  can,  of  course,  have  no  ratio  to  each  other. 
293 


.294  TIIGU  SCHOOL  ALGEBRA  [Ch.  XIX 

190.   Properties  of  ratios.     Since  ratios  are  quotients,  i.e.^ 
fractions,  therefore  they  have  all  the  properties  of  fractions. 
Thus,  a   ratio   is   not   changed    if   both   antecedent    and 
consequent  are  multiplied  or  divided  by  any  given  number. 
Again,  if  a,  ^,  and  k  are  positive,  and  a<h,  then  since 

a      a  +  k    .        (a—b)k  re  oo 

is  negative  (a  —  h  being  negative),  therefore  (§  185),  the  ratio 
a-\-k  '.  h  -{-k  is  greater  than  the  ratio  a  :  h.  [Translate  this 
important  fact  into  words  :  (1)  calling  a  ^  b  ix  proper  fraction ; 
and  (2)  calling  a-i-  b  a  ratio  less  than  unity.] 

EXERCISE  CXXVI 

What  is  the  ratio  of : 

1.  6  to  8?  3.    -10  to  24?  5.   -I  to  11? 

2.  50  to  15  ?  4.  6.3  to  .7  ?  6.  21  a;^  to  9  a;? 

7.  Which  term  of  a  ratio  corresponds  to  the  divisor  ?     What 
is  the  other  term  called  ?     Illustrate  from  Exs.  1-6. 

8.  Form  the  inverse  of  each  of  the  ratios  in  Exs.  1-6. 

9.  The  ratio  of  x  to  5  equals  2 ;  find  x,  and  check  your  work. 

10.  If  the  ratio  of  two  numbers  is  f  and  the  consequent  is 
6,  find  the  antecedent. 

In  each  of  Exs.  11-14,  find  (and  check)  the  value  of  x : 

11.  x^  :2  =  ^.  13.   64  :  a;  =  a;. 

12.  a;  +  5:  2a;  =  -7.  14.   25:a;2  =  9. 

15.  What  is  the  ratio  oi  x  to  y  when  7(x~y)=S(x-\-y)? 
when  af  -[-6y^  =  5xy? 

16.  A  yard  measure  is  divided  into  two  parts  whose  lengths 
are  in  the  ratio  7  :  11 ;  how  many  inches  in  each  part  ? 

17.  A  and  B  divide  $  100  between  them  so  that  A  receives  $13 
out  of  every  $  20.  What  is  the  ratio  of  A's  share  to  B's  share  ? 
How  many  dollars  does  each  receive  ? 


190-191]        llATIO,   PROPORTION,   AND    VARIATION  295 

18.  What  number  must  be  added  to  each  term  of  ^^  in  order 
that  the  resulting  ratio  shall  be  2:3?  Does  this  addition  in- 
crease or  diminish  the  given  ratio  ? 

19.  Which  is  the  greater  ratio,  5  +  2  :  17  +  2  or  5  :  17  ? 
21  +  8  :  11  -f  8  or  21 :  11  ?  Does  the  addition  of  the  same  positive 
number  to  both  terms  of  a  ratio  always  increase  the  latter's  value 
(cf.  §  190)?     Explain. 

20.  If  a,  b,  and  k  represent  positive  numbers,  translate  (1)  and 
(2)  below  into  words ;  then  give  three  numerical  illustrations  of 
each: 

(1)  ^^^  <  7  when  a<b', 

0  —  K         0 

(2)  ^'^>r  when  a>b. 
b  —k      b 

21.  By  a  method  similar  to  that  used  in  §  190,  show  the  cor- 
rectness of  (1)  and  (2),  Ex.  20. 

22.  If  X,  y,  and  z  are  positive  numbers,  which  is  the  greater 

,.     ,      -,     i     r>x   2ic4-5?/       ic-|-2?/o    x  —  y-\-z       x-\-y-\-Zo 
ratio  (and  why  ?), — — ^  or  —^ — ^  ?    ^L-L_  or     ^  ^  ^    ? 

2x-[-ly       x-\-3y      x-\-y  —  z       x  —  y  —  z 

23.  Show  that  the  following  ratios  are  all  equal:  $12  :  $9; 
8  bu.  oats  :  6  bu.  oats ;  4  T.  of  coal  :  3  T.  of  coal ;  10  in.  :  7^  in. ; 
4:3;  i:^^. 

24.  Eind  the  value. of  each  of  the  following  ratios:  4V2  :  V2; 
4V2:2;    7V3  in.  :  14V2  in.;     $5.80  :  29  cents. 

25.  Eind  two  integers  whose  ratio  equals  15|  :  9f .  Can  the 
ratio  of  any  two  numbers  whatever  be  expressed  as  the  ratio  of 
two  integers  (cf .  Ex.  24,  also  §  188)  ? 

26.  Which  of  the  pairs  of  numbers  (or  quantities)  in  Ex.  24 
are  commensurable  ?     Which  are  incommensurable  ?     Why  ? 

II.     PROPORTION 

191.  Definitions.  If  a,  5,  c,  and  d  are  any  four  numbers 
such  that  a  :  b  —  c  :  d,  then  these  numbers  are  said  to  be 
proportional,  or  to  form  a  proportion  ;  i.e.,  a  proportion  is 
a  statement  that  two  ratios  are  equal. 


296  HIGH  SCHOOL  ALGEBRA  [Ch.  XIX 

The  proportion  a  :b  =  e  :  d  (sometimes  written  a  :  b  ::  c  :  d^ 
is  read  :  "  the  ratio  oi  a  to  b  equals  the  ratio  of  c  to  c?,"  and 
also  "a  is  to  5  as  <?  is  to  c?."  In  this  proportion  a  and  d  are 
called  the  extremes,  while  b  and  e  are  called  tlie  means. 

Ji  a  :  b  =  e  :  d,  then  d  is  said  to  be  the  fourth  proportional 
to  a,  5,  and  c  ;  while  ii  a  :  b  =  b  :  c,  then  c  is  called  the  third 
proportional  to  a  and  5,  and  6  is  called  the  mean  proportional 
between  a  and  e. 

A  succession  of  equal  ratios,  in  which  the  consequent  of 
each  is  also  the  antecedent  of  the  next,  is  called  a  continued 
proportion  ;  thus  a  :  b  =  b  :  c  =  e  :  d  =  d  :  e  =  •  •  -^  is  a  continued 
proportion. 

EXERCISE   CXXVII 

1.  Is  it  true  that  8  :  12  =  10 :  15  ?  Why  ?  How  is  this  pro- 
portion read  ?     What  does  it  mean  ? 

2.  In  Ex.  1  name  the  means  and  the  extremes  of  the  propor- 
tion, also  the  fourth  proportional  to  8,  12,  and  10. 

3.  Is  it  true  that  8  :  10  =  12  :  15  ?  How  does  this  proportion 
compare  with  that  in  Ex.  1  ?  Does  a  proportion  remain  true 
after  its  means  have  been  interchanged  ?  Try  several  numerical 
examples  and  compare  Principle  4,  p.  298. 

4.  By  arranging  the  numbers  3,  4,  6,  and  8  in  different  ways, 
make  three  different  proportions. 

5.  Is  6  a  mean  proportional  between  4  and  9?  between  18 
and  2  ?  Is  the  same  thing  true  of  —  6  ?  Name  the  third 
proportional  in  each  case. 

6.  Show  that  2  :  6  =  6  :  18  =  18  :  54  =  54  :  162  is  a  continued 
proportion  in  which  each  ratio  equals  ^.  Write  a  continued 
proportion  of  five  ratios  each  of  which  equals  f. 

192.  Important  principles  of  proportion.  Since  a  propor- 
tion is  merely  an  equation  whose  members  are  fractions, 
therefore  the  principles  of  proportion  may  be  derived  from 
those  governing  equations  and  fractions. 


101-192]        RATIO,   PROPORTION,   AND    VARIATION  297 

Note.  Before  memorizing  the  following  principles  (1-8)  the  pupil  should 
illustrate  each  by  one  or  more  numerical  examples  ;  he  should  also  try  to 
invent  a  proof  of  his  own  for  each  principle,  before  reading  the  printed  proof. 

Principle  1.  If  four  numbers  are  in  proportion,  then  the 
product  of  the  extremes  equals  the  product  of  the  means. 

For,  let  a,  h,  c,  and  d  be  any  four  numbers  which  are  in 
proportion,  then  a  :h  =  c  :  d\ 

a      c 

whence  ad  =  he,         [clearing  of  fractions 

which  was  to  be  proved. 

Principle  2.  If  the  product  of  two  numbers  equals  the 
product  of  two  others,  then  either  pair  may  be  made  the 
extremes  of  a  proportion  in  which  the  other  pair  are  the  means. 

For,  if  ad  =  be, 

a      o 
then  7  ~  ;j'  [dividing  by  hd 

0      ct 

i.e.,  a  '.b  =  c  :  d. 

In  the  same  way  it  may  be  shown  that,  if  ad  =  he,  then 

h  :  a  =  d  :  c,    c  :  a  =  d  :  h,  etc. 

Eemark.  From  the  proof  just  given  it  follows  that  the  correct- 
ness of  a  proportion  is  established  when  it  is  shown  that  the  product 
of  the  7neans  equals  the  product  of  the  extremes  ;  this  test  is  very 
useful. 

Principle  3.  i/"  four  numbers  are  in  proportion,  then 
they  are  in  proportion  by  inversion  ;  i.e.,  the  second  is  to  the 
first  as  the  fourth  is  to  the  third. 

For,  ii  a  :  h  =  c  :  d,  then  ad  =  he  (why  ?)  ;  hence  h  :  a  =  d  :  c 
(cf.  Principle  2,  Remark). 

Suggestion.  Let  the  pupil  state  Principles  4-7  below  entirely  in  verbal 
language,  and  prove  each  in  detail  (cf.  statement  and  proof  of  Principle  2). 


298  IIJGH  SCHOOL  ALGEBRA  [Ch.  XIX 

Principle  4.  If  four  numhers  are  in  proportion^  then  they 
are  in  proportion  hy  alternation;  ^.e.,  if  a:h  —  c:d^  then 
a  :  c  =  b  :  d. 

Principle  5.  If  four  rmmbers  are  in  proportion^  then  they 
are  in  proportion  by  composition  ;  i.e.^  ii  a  :  b  =  c  :  d,  then 
(a  +  ^)  :  a=  (^c  -{-  d')  :  e  ;    [also,  (a  +  6)  :  5  =  (c  +  6?)  :  c?]. 

Principle  6.  If  four  numbers  are  in  proportion^  then  they 
are  in  proportion  by  division  or  separation  ;  i.e.,if  a  :  b  =  c  :  d, 
then  (a  —  b^  :  a=  (^c  —  d}  :  c;  [also,  (^a  —  b^  :  b=  (^c  —  d^  :  dj^. 

Principle  7.  If  four  numbers  are  in  proportion,  then 
they  are  in  proportion  by  composition  and  division  ;  i.e.^  if 
a  :  b  =  e  :  d,  then  (^a  +  5)  :  (a  —  6)  =  ((?  +  t?)  :  (c  —  c?). 

Principle  8.  In  a  series  of  equal  ratios  the  sum  of  the 
antecedents  is  to  the  sum  of  the  consequents  as  any  antecedent 
is  to  its  own  consequent. 

Thus,  if     a'.b  —  c:d=e:f=g:h  =  --'  =  x'.y., 

then  <ia  +  c  +  e+g-\--"  +  x):  (^b -j- d-hf+ h+ ■-■ -^  y}  =  e  :f. 

To  prove  this  theorem,  let  each  of  the  given  equal  ratios 
be  represented  by  a  single  letter,  say  r ; 

,  •,  a  c  e  a  -,  x 

then  -=r,    -=r,    -  =  r,    ^  =r,  •••,  and  -  =r, 

b  d  f  h  y 

hence      a  =  h\  c  =  dr^  e  =fr^  g=hr.,  •••,  and  x  =  yr^ 

and,  adding  these  equations,  member  to  member, 

a^c^e-\-g-\-"-^x  =  Q)^d^f-\-h-\--'^y)r, 

and  therefore         ^  +  ^  + ^ +.^+ •>• +^  ^^^e 
b+d+f-\-h^--  +  y  f 

which  proves  the  principle. 

Note.  As  in  the  proof  just  given,  so  it  will  often  be  found  advantageous 
to  represent  a  ratio  by  a  single  letter. 


192]  RATIO,   PROPORTION,   AND    VARIATION  299 


EXERCISE  CXXVIII 

Using  Prin(3iple  1,  find  .*;  in  each  of  Kxs.  1-4  : 

1.  14  :  3  =  56  :x.  3.    -  16  :  .^•  =  18  :  7. 

2.  aj :  -  5  =  20  :  -  2.  4.    J  :  ic  =  x  :  g^^  . 

Find  a  mean  proportional  between  each  of  the  following  pairs 
of  numbers  (cf.  Ex.  4)  : 

5.  3,  27.  7.   25,  —  4.  -  9.    ain'^,  a^m. 

6.  -2,-5.  8.    .25,  .09.  10.   a  +  h,a-b. 

11.  How  many  answers  has  each  of  Exs.  5-10  ?  Why  ? 
Show  that  the  mean  proportional  between  any  two  numbers 
equals  the  square  root  of  their  product. 

12.  Find  the  third  proportional  to  1  and  4 ;  to  —  25  and  —  40 ; 
also  the  fourth  proportional  to  m  —  n,  m^  —  rr,  and  m  +  n. 

13.  Using  Principles  2-7,  make  seven  different  proportions 
from  the  equation  cd  =  mn. 

14.  Add  1  to  each  member  of  the  equation  a:b  =  c:  d;  write 
the  result  as  a  proportion  and  thus  prove  Principle  5. 

15.  Prove  that  like  powers  (also  like  roots)  of  proportional 
numbers  are  proportional,  i.e.,  prove  that  if  a:b  =  G:  d,  then 
a"  :  ft**  =  c"  :  d\ 

16.  If  a:  b  =  c:  d  and  e:f  =  g:h,  show  that  ae  :bf=  eg:  dh ; 
also  translate  this  principle  into  verbal  language. 

17.  li  a:  b  =  c:d  and  e  : /=  a  :  h,  show  that  -  :  -  =  -  : -• 

e    f     g   h 

Hint.     Use  a  single  letter  to  represent  a  ratio  (cf.  proof  of  Principle  8). 

li  p :  q  =  r :  s,  Siud  m  and  n  are  any  numbers  whatever,  show 
that  the  proportions  in  Exs.  18-25  are  true. 

18.  mp  :  7iq=mr :  ??s(cf.  §  190).        21.   s^:  q^  =  r'^ :  p^ 

19.  op:r  =  5q:s.  22.  p -\-q  :2  p  =  r  +  s  :  2  r. 

20    r:s=-:     '  23.  pr  :  qs=  r^ :  s\ 

q  p 

24.  j9'^-4r-:^"-4s'»=-|":  -^• 

HIGH    ^ClI.    ALG. — 20 


300  HIGH  SCHOOL  ALGEBRA  [Ch.  XIX 


25.  2)  +  q:  r+  s  =  Vp~  -f  g^  ■  V?^T^. 

26.  li  a:b  =  c:  d  =  e  :f=g  :  h=  •",  and  I,  m,  7i,  p,  -"  are  any 
numbers  whatever,  show  that 

(ma -\- Ic  —  ne -i- pg -^  •••)  -.{mb  -\- Id  —  nf -\- ph -{ )  =  a:h. 

Hint.     Compare  §  190,  also  Principle  8. 

27.  If  {p-\-q^-r+s){p—q—r-{-s)  =  {p-q^r-s){p  +  q—r—S), 
show  that  p  :  q  =  r  :  s. 

28.  If   ax  +  cy^ay  +  cz^az  +  cx^    ^^^^   ^^^^.    ^^^,^   ^j   jj^^^^ 

?>.?/  -|-  dz      bz  +dx      bx+  dy 

ratios  equals  ^LjL^  (cf.  Principle  8). 
b  -\-d 

By  the  principles  of  proportion  solve  the  equations  : 

29.  a;:15  =  aj-l:12.  33.   a: :  27  =  2/ :  9  =  2  :  a; -?/. 


30.  a;-:32  =  a;  +  2:12.  a;+Va;-l      13 

34.    ^=iiiz  =  — 

31.  {\cx+V)rdx'=\c\  x--Jx-l       7 


'x-y=2, 


Hint.     Apply  Principle  7. 


32.    \  Q^-^f  ^5  32     Va;  +  7  + V^^4  + Va?^ 

(a?  +  2/f     9  ■    Va;  +  7  -  V^     4  —  Vx 

36.  AVhat  number  must  be  added  to  each  of  the  numbers  7,  9, 
11,  and  21  in  order  that  the  four  sums  may  be  proportional  ? 

37.  In  the  triangle  ABC^  ^/^  divides  BC  into  two  parts,  5^ and 
KC^  respectively  proportional  to  AB  and  AC.  If  ^J5  =  10  in., 
AC  =  16  in.,  and  5(7=  20  in.,  find  BK and  KC  (Draw  a  figure 
to  illustrate.) 

38.  Find  two  different  numbers,  m  and  w,  such  that 

m-\-n\m  —  n\  m^  +  n^  =  5:3:  51.* 

39.  The  perimeter  of  a  triangle  whose  sides  are  in  the  ratio 
5  :  6  :  8,  is  57  meters ;  find  the  lengths  of  the  sides. 

40.  How  may  $10  be  divided  among  three  boys  so  that  for 
every  dollar  the  first  receives,  the  second  shall  receive  15  cents 
and  the  third  10  cents  ? 


*  The  expression  a:h:c  =  x:y'.z,  means  that  a  :  h  ■=  x  :  y,  a  :  c  =  x  :  z^ 
and  b  :c  =  y  :  z;  and  also  the  equivalent  statement  a  :  x  =  b :  y  =  c  :  z. 


192-194]        RATIO,   PROPORTION,   AND    VARIATION  301 

41.  Two  rectangles  are  equal  in  area.  If  their  widths  are  as 
2  :  3,  find  the  ratio  of  their  lengths. 

42.  The  sides  of  a  certain  rectangle  are  in  the  ratio  7  : 3. 
Compare  the  area  of  the  rectangle  with  that  of  a  square  which 
has  the  same  perimeter. 

43.  If  a:b,  c :  d,  e\f,  g :  h,  •••  are  unequal  ratios,  in  which 
a,  b,  c,  '"  are  positive  numbers,  and  if  a:  b  is  the  greatest  and 
e  :  /  the  least  among  these  ratios,  show  that 

is  less  than  a  :  b,  but  greater  than  e  :  /  (cf.  proof  of  Principle  8). 

III.     VARIATION 

193.  Variables,  constants,  and  limits.  Many  questions  in 
mathematics  are  concerned  with  numbers  whose  values  are 
changing  ;  such  numbers  are  usually  called  variables,  while 
numbers  whose  values  do  not  change  are  called  constants. 

If  the  difference  between  a  variable  (in  the  course  of  its 
changes)  and  a  constant  may  become  and  remain  less  than 
any  assigned  number  however  small,  then  this  constant  is 
said  to  be  the  limit  of  the  variable. 

Thus,  your  own  age,  the  height  of  the  mercury  column  in  a  thermometer 
tube,  the  length  of  the  shadow  cast  by  a  given  flagstaff,  etc.,  are  variables  ; 
while  the  difference  between  the  ages  of  two  given  men,  the  weight  of  the 
mercury  in  a  given  thermometer,  the  length  of  a  certain  flagstaff,  etc.,  are 
constants. 

Again,  the  decimal  .3333  •••  (i.e.,  .3  +  .03  +  .003  H — )  is  a  variable  whose 
limit  is  I ;  this  decimal  grows  larger  and  larger  as  more  and  more  places  are 
included,  and  may  thus  be  made  to  differ  from  i  by  less  than  any  assigned 
number  however  small. 

So,  too,  1  +  J  +  I  +  I  +  Jg  •••  is  a  variable  whose  limit  is  2  (cf.  §  202). 

n      n  0 

194.  Interpretation  of  the  forms  -,  -,  and  -•    Two  of  these 

0    oo  0 

forms  were  first  hiet  with  in  §  41,  and  were  there  inter- 
preted by  assuming  that  the  definition  of  division,  given  in 
§  8,  remains  valid  for  infinitely  large  numbers  and  for  zero. 


302  HIGH  SCHOOL  ALGEBRA  [Ch.  XIX 

It   is   better,   however,  to   interpret  these  forms  from  the 
standpoint  of  variables  and  limits. 

(i)  If   the  values   1,  -jIq-,  y^^,  -f^foo^,   •••,  are  successively 

assigned   to   x^  what   are   the  corresponding  values   of   -? 

of  -,  where  a  is  any  finite  constant  whatever  ?     Answer  the 

same  questions  when  x  takes  the  successive  values  1,  10, 
102,  103,  .... 

These  examples  illustrate  two  important  facts,  viz. : 

(1)  ^8  the  divisor  grows  smaller  and  smaller^  approaching 
zero  as  a  limit  (the  dividend  beirig  a  finite  constarit^^  the  quotient 
increases  without  limit. 

(2)  As  the  divisor  increases  without  limit  (the  dividend  being 
a  finite  constant^ ^  the  quotient  approaches  zero  as  a  limit. 

For  the  sake  of  brevity,  (1)  and  (2)  are  often  expressed 
by  the  equations 

-  =  00  and  —  =  0, 

0  CX) 

respectively ;  but  the  interpretation  of  these  equations  is  as 
stated  in  (1)  and  (2)  above. 

(ii)  In   the  fraction   -— ,  as  x  takes   the  successive 

x—1 

values  1.1,  1.01,  1.001,  1.0001,  what  limit  is  approached  by 

the  numerator?   by  the  denominator?   by  the  value  of  the 

fraction?      Answer   the   same   questions   for    — -,    as   x 

approaches  2  as  a  limit. 

These  examples  illustrate  the  fact  that  as  a  dividend  and 
its  divisor  each  approach  zero  as  a  limit,  the  quotient  may 
approach  any  value  whatever  ;  this  is  often  expressed  by 
saying  that 

-  is  indeterminate. 


194]  RATIO,   PROPORTION,  AND    VARIATION  303 


EXERCISE   CXXIX 

1.  Which  of  the  following  quantities  are  constants  and  which 
are  variables :  (1)  the  circumference  of  a  growing  orange  ?  (2)  the 
length  of  the  shadow  cast  by  a  certain  church  steeple  between 
sunrise  and  sunset?  (3)  the  length  of  the  steeple  itself?  (4)  the 
time  since  the  discovery  of  America  ?  (5)  the  interest  earned  by 
an  outstanding  note  ?  (G)  the  principal  of  the  note  ? 

2.  A  point  P  moves  through  half  the  distance  AB  {i.e.,  to  P'), 
then  through  half  the  remaining  distance  p  pn  pu 

(i.e.,  to  P"),  then  through  half  the  remain-        i        i    i 

ing  distance  (i.e.,  to  P"),  and  so  on.     Show    ^ 

that  the  distance  from  A  to  P  is  a  variable  whose  limit  is  AB. 

3.  In  Ex.  2,  is  the  distance  from  P  to  J5  a  constant  or  a 
variable  ?     What  is  its  limit  ?     Explain  both  answers. 

4.  If  X  takes  in  succession  the  values  .6,  .06,  .006,  .0006,  •••, 
what  is  its  limit  ?     Why  ? 

5.  What  is  the  limit  of  the  variable  sum  .6  +  .06  -f-  .006  -\ 

(i.e.,  of  the  decimal  .6666 ••-)  ?     Explain. 

6.  If  r  is  any  finite  constant,  trace  the  changes  in  the  quotient 
r/s  as  s  passes  through  the  values,  3,  1,  ^,  ^,  -^j,  •••;  also  as  s 
passes  through  the  values  3,  9,  27,  81,  •••.  Is  there  a  limit  to 
the  quotient  in  the  first  case  ?  in  the  second  ?     Explain. 

7.  Translate  into  verbal  language  [cf .  §  194  (i)] : 

(1)  ^  =  00;  (2)  ^=0. 

8.  As  X  approaches  the  limit  1,  what  limit  does approach 

X  —1 

in  form  f  in  value  f    Answer  the  same  questions  for  the  fractions 
a;  —  1       x^  —  x  1  3  a^  — 2a;  — 1 


a^-1'     a;-l' 


ar 


9.   By  means  of  your  answers  to  the  questions  in  Ex.  8,  illus- 
trate the  fact  that  -  is  indeterminate  in  value. 


304  IlIGIl  SCHOOL  ALGEBRA  [Cii.  XIX 

195.  Direct  and  inverse  variation;  etc.  Of  two  variables 
which  are  so  rehited  that,  during  all  their  changes,  their  ratio 
remains  constant,  each  is  said  to  vary  (also  to  vary  directly) 
as  the  other.  The  symbol  employed  to  denote  variation  is  ^ ; 
it  stands  for  the  words  "  varies  as,"  and  the  expression  acch 
is  read  '•^  a  varies  as  5." 

If  a  Qc  ^,  ^.g.,  if  a  :  6  =  A;,  a  constant,  then  a  =  kb  (why  ?)  ; 
hence  a  variation  statement  may  be  converted  into  an 
equation. 

E.g.,  if  a  tank  contains  v  cii.  ft.  of  water,  each  cubic  foot  weighing  02.5  lb., 
and  if  the  total  weight  of  the  water  is  to  lb.,  then  : 

(1)  When  V  changes  (as  it  must,  for  example,  while  the  tank  is  filling), 
w  changes  also. 

(2)  Since,  no  matter  how  the  quantity  of  water  changes,  to  =  62.5 -w,  or 
w  :  V  =  62.d,  therefore  wcav;  i.e.,  the  weight  of  watisr  varies  as  its  volume. 

One  of  two  numbers  is  said  to  vary  inversely  as  the  other 
if  the  ratio  of  the  first  to  the  reciprocal  of  the  second  is 
constant.  If  a  varies  inversely  as  5,  then  a  '  h  =  k:  let  pupils 
fully  explain  why. 

Again,  if  x,  y,  and  z  are  variables  such  that  x  =  ki/z^  where 
k  is  Si  constant,  then  x  is  said  to  vary  jointly  as  ^  and  z ;  and 

if  a;  =  -^,  then  x  is  said  to  vary  directly  as  y  and  inversely  as  z. 

z 

E.g.,  the  time  required  for  a  railway  train  to  travel  a  given  distance  varies 
inversely  as  the  speed  ;  for  if  t,  r,  and  d  represent,  respectively,  the  time,  rate, 

and  distance,  then  t  'r  =  d,  i.e.,  t:  ~  =d,  where  d  is  constant. 

r 

Again,  the  cost  of  a  railway  journey  varies  jointly  as  its  length  and  its 
cost  per  mile  ;  while  the  number  of  posts  required  to  build  a  certain  fence 
varies  directly  as  the  length  of  the  fence,  and  inversely  as  the  distance  be- 
tween the  posts. 

Note.  It  should  be  remarked  in  passing  that  such  an  expression  as  wccv 
above  (i.e.,  the  weight  of  water  varies  as  its  volume)  is  merely  an  abbreviated 
form  of  the  proportion 

w:w'  =v:v', 

wherein  w  and  w'  stand  for  the  respective  weights,  and  v  and  v'  for  the 
volumes,  of  any  two  quantities  of  water. 


195]  RATIO,   PROPORTION,   AND   VARIATION  305 

The  theory  of  variation  is,  therefore,  substantially  included  in  that  of 
ratio  and  proportion,  and  the  only  reason  for  even  defining  the  expressions 
"varies  as,"  "varies  inversely  as,"  etc.,  here,  is  that  this  convenient 
phraseology  is  so  well  established  in  physics,  chemistry,  etc. 


EXERCISE   CXXX 

1.  Explain  and  illustrate  the  following  statements : 

(1)  The  interest  earned  by  a  certain  principal  varies  as  the  time. 

(2)  The  circumference  of  a  circle  varies  as  its  radius. 

2.  State  (1)  and  (2)  of  Ex.  1  as  equations  (cf.  §  195),  also  as 
proportions  (cf.  §  195,  Note). 

3.  It  xccy  and  if  a?  =  12  when  y  =  3,  find  the  equation  connect- 
ing X  and  y ;  also  find  x  when  y==7. 

Solution.  Since  xccy,  therefore  x  =  ky  where  k  is  constant  (why?)  ; 
moreover,  when  x  =  12  and  ?/  =  3,  the  equation  x  =  ky  gives  k  =  4.  There- 
fore, under  the  given  conditions,  x  =  4y;  hence,  when  y  =  7,  x  =  28. 

4.  li  aocb  and  if  a  =  89  when  b  =  —3,  find  a  when  6  =  2;  also 
when  &  =  I ;  also  find  b  when  a  =  —  65. 

5.  If  ^  X  -B  and  BccC,  show  that  AccC. 
Hint.     Show  that  A  =  kC\  where  k  is  some  constant. 

6.  If  m  cc  n  and  pccriy  prove  that  m±pccn. 

7.  If  3  m^— 18  oc  2n  + 1,  and  if  m  =  4  when  n  =  2,  find  m  when 
n  =  23.5. 

8.  The  area  of  a  circle  varies  as  the  square  of  its  radius.  If 
a  circle  whose  radius  is  10  ft.  contains  314.16  sq.  ft.,  find  the 
area  of  a  circle  whose  radius  is  5  ft. ;  of  one  whose  radius  is  12  ft. 

9.  Find  the  radius  of  a  circle  whose  area  is  twice  that  of  a 
circle  10  ft.  in  radius  (cf.  Ex.  8). 

10.  If  X  varies  inversely  as  y,  how  is  the  value  of  x  aifected  if 
y  is  doubled  ?  if  ?/  is  multiplied  by  10  ?  ii  y  is  divided  by  —  6  ? 
Explain. 

11.  Give  three  numerical  examples  of  inverse  variation. 


306  HIGH  SCHOOL  ALGEBRA  [Cii.  XIX 

12.  If  X  varies  inversely  as  y,  show  that : 

(1)  xy  =  k  (where  k  is  constant). 

(2)  x'  :x"  =  y"  :  y'  (where  x'  and  y',  x"  and  y"  are  correspond- 
ing vahies  of  the  variables). 

13.  If  x  varies  inversely  as  y,  and  if  x  =  4  when  y  =  2,  find  y 
when  x  =  —  8;  when  x  =  l^  ;  when  x  =  2.5. 

14.  If  x  varies  directly  as  y  and  inversely  as  z,  and  if  a;  =  —  12 
when  2/  =  2  and  z  =  7,  find  y  when  x  =  2  and  2  =  3. 

15.  Solve  Ex.  13  by  drawing  the  graph  of  the  equation  con- 
necting X  and  y  (cf.  §  141),  and  then  measuring  the  ^/-coordinates 
of  the  points  whose  respective  .^-coordinates  are  —  8,  1^,  and  2.5. 
Also  show,  from  the  graph,  that  any  change  in  x  makes  an  oppo- 
site change  in  y. 

16.  If  the  volume  of  a  pyramid  varies  jointly  as  its  base  and 
altitude,  and  if  the  volume  is  20  cu.  in.  when  the  base  is  12  sq.  in. 
and  the  altitude  is  5  in.,  what  is  the  altitude  of  the  pyramid 
whose  base  is  48  sq.  in.  and  whose  volume  is  76  cu.  in.  ? 

17.  The  distance  (in  feet)  through  which  a  body  falls  from  a 
position  of  rest,  varies  as  the  square  of  the  time  (in  seconds) 
during  which  it  falls.  If  a  body  falls  257^  ft.  in  4  sec,  how  far 
will  it  fall  in  5  sec.  ?  how  far  during  the  5th  second  ?  how  far 
during  the  7th  second  ? 

18.  If  the  intensity  of  light  varies  inversely  as  the  square  of 
the  distance  from  the  source  of  light,  how  much  farther  from  a 
lamp  must  a  book,  which  is  now  2  ft.  away,  be  removed  so  as  to 
receive  just  one  third  as  much  light  ? 

19.  The  weight  of  a  body  comparatively  near  the  earth's  sur- 
face varies  inversely  as  the  square  of  its  distance  from  the  earth's 
center.  Assuming  that  the  radius  of  the  earth  is  4000  mi.,  find 
the  weight  of  a  4-lb.  brick  2000  mi.  from  the  earth's  surface. 
(Two  solutions.) 

20.  The  number  of  oscillations  made  by  a  pendulum  in  a  given 
time  varies  inversely  as  the  square  root  of  its  length.  If  a  pen- 
dulum 39.1  inches  long  oscillates  once  a  second,  what  is  the 
length  of  a  pendulum  that  oscillates  twice  a  second? 


CHAPTER   XX 
SERIES  —  THE  PROGRESSIONS 

196.  Definitions.  A  series  is  a  succession  of  numbers 
which  proceed  according  to  some  definite  law.  The  num- 
bers which  constitute  the  series  are  called  its  terms. 

E.g.^  in  the  series  1,  2,  3,  5,  8,  13,  eacli  term  after  the  second  is  the  sum 
of  the  two  preceding  terms ;  in  the  series  2,  6,  18,  54,  162,  each  term  after 
the  first  is  3  times  the  preceding  term  ;  and  in  the  series  1,  4,  9,  16,  ••.,  81, 
each  term  is  the  square  of  the  number  of  its  place  in  the  series. 

A  series  which  consists  of  an  unlimited  number  of  terms 
is  called  an  infinite  series  ;  otherwise  it  is  a  finite  series. 

The  present  chapter  considers  only  the  simplest  kinds  of 
series — the  so-called  "progressions." 

I.    ARITHMETICAL  PROGRESSION 

197.  Definitions  and  notation.  An  arithmetical  series,  or 
arithmetical  progression  (designated  by  A.  P.),  is  a  series 
in  which  the  difference  found  by  subtracting  a  term  from 
the  next  following  term  is  the  same  throughout  the  series. 
This  constant  difference,  whether  positive  or  negative,  inte- 
gral or  fractional,  is  known  as  the  common  difference  of  the 
series. 

E.g.^  the  series  2,  5,  8,  11,  14,  ••.  is  an  A.  P.  whose  common  difference  is 
3.  So,  too,  the  series  18,  11,  4,  —  3,  —  10,  is  an  A.  P.  whose  common  differ- 
ence is  —  7. 

The  elements  of  any  given  A.  P.  are  the  first  term  (desig- 
nated by  a),  the  last  term  (Z),  the  common  difference  (c?), 
the  number  of  terms  (n),  and  the  sum  of  all  the  terms  (s). 

Thus,  in  the  series  2,  5,  8,  •••,  32,  the  elements  are  «  =  2,  Z  =  32,  (?=  3, 
11  =  11,  s=  187. 

307 


308  HIGH  SCHOOL  ALGEBRA  [Ch.  XX 


EXERCISE  CXXXI 

1.  Does  a  row  of  numbers  written  down  at  random  constitute 
a  series  ?     Explain. 

2.  Show  that  1,  7,  13,  19,  25  is  an  A.  P.  What  are  its  ele- 
ments ? 

3.  What  is  d  in  the  A.  P.  7,  11,  15,  19  ?  Extend  this  series 
four  terms  to  the  right ;  also  three  terms  to  the  left. 

4.  If  the  1st,  3d,  and  5th  terms  of  an  A.  P.  are  18,  24,  and  30, 
respectively,  find  d  and  write  8  consecutive  terms  of  the  series. 

5.  Write  10  consecutive  terms  of  the  series  in  which  19,  9, 
and  4  are  the  1st,  5th,  and  7th  terms,  respectively. 

6.  What  are  the  elements  of  the  A.  P.  5,  5  +  3,  5  +  6,  5  +  9, 
5  +  12  ?  How  is  any  term  of  this  series  formed  from  the  pre- 
ceding term  ? 

7.  Show  that  x,x+y,  x-\-2y,  x-{-3y,  •••  is  an  A.  P.  What 
is  d  in  this  series  ?  How  many  times  must  d  be  added  to  the 
first  term  to  make  the  2d  term  ?  to  make  the  3d  term  ?  the  7th 
term  ?    the  10th  term  ?    the  nth  term  ? 

8.  Show  from  the  definition  of  an  A.  P.  that  such  a  series 
may  be  written  in  the  form 

a,    a+d,    a-j-2d,    a  +  3d,  '•-,  I —  2d,    l—d,    I, 
wherein  a,  d,  and  I  represent,  respectively,  the  first  term,  com- 
mon difference,  and  last  term. 

198.  Formulas.  The  elements  of  an  A.  P.  are  connected 
by  the  two  fundamental  equations  (formulas)  numbered  (1) 
and  (2)  below. 

Since  each  term  of  an  A.  P.  may  be  derived  by  adding  d 
to  the  preceding  term  (cf.  Exs.  6-8,  above),  therefore,  if 
I  stands  for  the  nth  term 

l  =  a-}-(n-l)d.  (1) 


107-198]  SERIES  — THE  PPiOGnESSIONS  309 

Again,  since  the  sum  of  the  terms  of  an  A.  P.  may  be 
written  in  each  of  the  two  following  forms, 

s  =  a-^  (a-\-d)-^(a-\-2d)-h--'  +  (I- 2  d)  -h  (I- d) -^  I 
and  s  =  I  i-  {I  -  d)  -{-  {I  -  2  d)  -\-  •"  +  (a  -^  2  d)  -{-  {a  -\-  d)  -^  a, 
therefore,  by  adding  these  equations,  term  by  term, 

i.e.^  2s  =  n(^a-\-  Z),  [n  terms 

whence  *  =  -^-^ — ^ ;  (2) 

or,  substituting  the  value  of  I  from  (1), 

n[Za  +  (n-l)<n, 
"-^  2 

Note.  If  any  three  of  the  five  elements  of  an  A.  P.  are  given,  the  other 
two  can  always  be  found  from  (1)  and  (2)  above,  because,  in  that  case,  the 
two  unknown  elements  will  be  connected  by  two  independent  equations. 

Ex.  1.     Find  the  sum  of  8  terms  of  the  A.  P.  —  3,  —  1,  1,  3  •••. 

Solution.     Here  a=— 3,  d  =  2,  w  =  8; 

whence,  from  (1),  Z  =  -3  +  (8  - 1)  2  =  11, 

and  from  (2),  s  =  ^(-^  +  ^^)  =  32 ; 

Li 

i.e.,  the  sum  of  8  terms  of  the  A.  P.  is  32. 


EXERCISE  CXXXII 

Verify  formulas  (1)  and  (2)  above  for  the  following  series : 

2.  10,  13,  16,  19,  22,  25,  28. 

3.  26,  19,  12,  5,  -2,-9,  -16,  -23,  -30. 

4.  _8,-5|,  -3i,  -1,H,3|,6. 

5.  By  means  of  §  198  (1),  find  the  17th  term  of  7,  11,  15,  ... ; 
then  by  §  198  (2),  and  without  writing  all  the  terms,  find  the  sum 
of  the  first  17  terms  of  this  series. 

6.  As  in  Ex.  5,  find  the  12th  term  of  1,  3.5,  6,  8.5,  •..;  also 
the  sum  of  the  first  8  terms. 


310         •  HIGH  SCHOOL  ALGEBRA  [Ch.  XX 

Find  the  sum  of : 

7.  Ten  terms  of  4,  11,  18,  .... 

8.  Thirty  terms  of  -  2,  -  0.5,  1,  2.5,  .... 

9.  Nineteen  terms  of  2,  5,  8,  •.-. 

10.  k  terms  of  2,  5,  8,  •••. 

11.  n  terms  of  5,  5  -f  A',  5  +  2  A:,  5  -f  3  A;,  • ... 

12.  t  terms  of  li,  2  h,  3h,  •••. 

13.  Find  the  sum  of  the  even  numbers  from  2  to  100  inclusive. 
Compare  your  result  with  that  in  Ex.  12  when  h  =  2  and  t  =  50. 

14.  How  many  strokes  are  made  in  a  day  (24  hours)  by  a  clock 
which  strikes  the  hours  only  ? 

15.  Suppose  that  50  eggs  are  placed  in  a  row,  each  2  yd.  from 
the  next^  and  a  basket  2  yd.  beyond  the  last  egg ;  how  far  would 
a  boy,  starting  at  the  basket,  walk  in  picking  up  these  eggs  and 
carrying  them,  one  at  a  time,  to  the  basket  ? 

16.  If  a  body  falls  16.1  feet  during  the  first  second,  3  times  as 
far  during  the  next  second,  5  times  as  far  during  the  third  second, 
etc.,  how  far  will  it  fall  daring  the  8th  second?  how  far  during 
the  first  8  seconds  ? 

17.  By  means  of  §  198  (1)  find  n  when  a  =  2,  cZ  =  4,  1  =  66', 
also  find  s  for  this  series. 

18.  If  a  =  -10,d  =  3,  s  =  35,  findZandn. 

Hint.  Substitute  in  §  198  (1)  and  (2)  and  solve  the  resulting  equations 
for  I  and  n  (cf.  §  198,  Note). 

19.  If  a  =  l,  d  =  -|,  71  =  9,  findZand.9. 

20.  If  1  =  -^,  n  =  lS,  s  =  -45i,  find  a  and  d. 

21.  How  many  consecutive  odd  integers  (beginning  with  1) 
must  be  added  to  give  the  sum  225  ?    441  ?     (Cf.  Ex.  18.) 

22.  If  s  =  112  and  n  =  7,  determine  the  unknown  elements  in 
the  series  ..«,  10,  13,  16,  .-.,  and  write  the  series. 

23.  If  s,  n,  and  d  are  given,  find  a  and  I;  i.e.,  find  a  and  I  in 
terms  of  s,  n,  and  d  (cf.  Ex.  22). 


198-191)]  SERIES—-  THE  PEOGliESSIONS  311 

24.  Find  a  and  ?i  in  terms  of  d,  I,  and  s.  Make  up  and  solve 
eight  other  examples  of  this  kind. 

25.  Show  that  an  A.  P.  is  fully  determined  when  any  three  of 
its  elements  are  given. 

26.  If  the  6tli  and   11th   terms   of   an  A.  P.  are   17   and   32, 

respectively,    find  the  common  difference,  and   also  the  sum  of 

the  first  11  terms. 

Hint.  Since  the  6tli  term  is  17,  therefore  17  =  a  +  5  (Z.  Similarly, 
32  =  a  +  10  d 

27.  Show  that  if  each  term  of  an  A.  P.  is  multiplied  (or 
divided)  by  any  given  number,  the  resulting  products  (or 
quotients)  are  themselves  in  arithmetical  progression. 

28.  If  each  term  of  an  A.  P.  is  increased  or  diminished  by  any 
given  number,  will  the  results  be  in  arithmetical  progression  ? 
Explain. 

199.  Arithmetical  means.  The  two  end  terms  of  an  A.  P. 
are  called  its  extremes,  while  all  the  other  terms  are  called 
arithmetical  means  between  these  two. 

E.g.,  in  the  A.  P.  5,  9,  13,  17,  21,  between  the  extremes  5  and  21  there 
are  3  arithmetical  means  (viz.,  9,  13,  and  17). 

In  an  A.  P.  of  3  terms  the  (one)  arithmetical  mean 
between  the  extremes  equals  half  their  sum  ;  for  if  A  is 
the  arithmetical  mean  between  a  and  5,  then 

A  —  a  =  h  —  A^      [definition  of  an  A.  P. 

whence  4  =?-LA.  % 

Ex.  1.     Find  the  arithmetical  mean  between  3  and  27. 
Solution.     The  arithmetical  mean  between  3  and  27 

=  3  +  27^  15_ 

2 

Ex.  2.     Insert  5  arithmetical  means  between  3  and  27. 

Solution.  In  this  series  a  =  3,  1  =  27,  and  (since  there  are  to 
be  5  means)  n  =  5-}-2  =  7;  whence,  from  §  198  (1),  d  =  4,  and 
the  required  series  is  3,  7,  11,  15,  19,  23,  27. 


312  HIGH  SCHOOL   ALGEBRA  [Ch.  XX 

EXERCISE  CXXXIII 

3.  Find  the  arithmetical  mean  between  14  and  9;  between 
-5  and  17;    -3  and  -4;   fandf;    -  2.75  and  11.4. 

4.  Insert  4  arithmetical  means  between  12  and  27. 

5.  Insert  15  arithmetical  means  between  19  and  131. 

6.  Insert  20  arithmetical  means  between  16  and  —  40. 

7.  Insert  12  arithmetical  means  between  —  f  and  f . 

8.  Insert  7  arithmetical  means  between  —  .08  and  —  .0032. 

9.  If  m  arithmetical  means  are  inserted  between  two  given 
numbers,  such  as  a  and  b,  show  that  the  common  difference  of 
the  series  thus  formed  is  (6—  a)  -j-  (m  -f- 1). 

10.  What  does  the  formula  of  Ex.  9  become  when  m  =  1  ?  Is 
this  consistent  with  the  formula  for  A  obtained  in  §  199  ? 

11.  Without  actually  finding  the  means  asked  for  in  Ex.  2, 
find  the  sum  of  the  series  formed  by  inserting  them. 

12.  Find  three  numbers  in  A.  P.  whose  sum  is  15,  and  the  sum 
of  whose  squares  is  107. 

Hint.     Let  x  —  y,  x,  and  x  -{-  y  represent  the  required  numbers. 

13.  Find  three  numbers  in  A*.  P.  whose  sum  is  18  and  whose 
product  is  202i. 

14.  The  sum  of  the  first  seven  terms  of  an  A.  P.  is  105,  and  the 
sum  of  the  third  and  fifth  terms  is  10  times  the  first  term.  Find 
the  series. 

15.  The  product  of  the  extremes  of  an  A.  P.  of  three  terms  is  4 
less  than  the  square  of  the  mean,  and  the  sum  of  the  series  is  24. 
Find  the  series. 

16.  The  sum  of  four  numbers  in  A.  P.  is  14,  and  the  product 
of  the  means  is  12.     What  are  the  numbers  ? 

Hint.     Let  x  —  3y,x—y,x  +  y,x-\-Sy  represent  the  series. 

17.  The  sum  of  an  A.  P.  of  five  terms  is  15,  and  the  product  of 
the  extremes  is  3  less  than  the  product  of  the  second  and  fourth 
terms.     Find  the  series. 

18.  How  many  arithmetical  means  must  be  inserted  between  4 
and  25  so  that  the  sum  of  the  series  may  be  116  ? 


199-200]  SERIES— THE  PROGRESSIONS  813 

19.  A  number,  expressed  by  three  digits  in  A.  P.,  equals  30.4 
times  the  siim  of  its  digits ;  but  if  9  is  added  to  the  number,  the 
units'  and  tens'  digits  will  be  interchanged.     Find  the  number. 

20.  In  the  series  1,  3,  5,  ••  •,  what  is  the  n\h.  term  ?  Prove  that 
the  sum  of  the  first  n  odd  numbers,  beginning  with  1,  is  nl 

II.    GEOMETRIC  PROGRESSION 

200.  Definitions  and  notation.  A  geometric  series,  or  geo- 
metrical progression  (designated  by  G.  P.),  is  a  series  in 
which  the  quotient  of  each  term  (after  the  first)  divided  by 
the  next  preceding  term  is  the  same  throughout  the  series. 
This  constant  quotient  is  called  the  common  ratio,  or  simply 
the  ratio,  of  the  series. 

E.g.^  the  numbers  2,  6,  18,  54,  •••  form  a  geometric  series  whose  ratio  is  3 ; 
while  |,  —  1,  f ,  -  I,  V",  •••  is  a  G.  P.  whose  ratio  is  -  |. 

The  elements  of  any  given  G.  P.  are  the  first  term  (a), 
the  last  term  (Z),  the  number  of  terms  (n)^  the  ratio  (r), 
and  the  sum  of  all  terms  (s). 

E.g.,  in  the  G.  P.  2,  -6,  18,  -54,  162,  -486,  1458,  a  =  2,  Z  =  1458, 
n  =  7,  r  =  -  3,  and  s  =  1094. 

EXERCISE  CXXXIV 

Which  of  the  following  are  geometric  series  ?    Explain. 

1.  7,  21,  63,  189,  567. 

2.  1,  4,  16,  64,  192,  576. 

3.  -6,  12,  -24,  48,  -96,  192,  -384. 

4.  What  are  the  elements  of  the  G.  P.  in  Ex.  3  ?  Extend  this 
series  two  terms  to  the  right,  also  five  terms  to  the  left. 

5.  If  a  is  the  first  term  of  a  G.  P.,  and  r  the  ratio,  what  is  the 
second  term?  the  third?  the  fourth?  the  fifth?  the  fourteenth? 
the  twenty-third  ?  the  nth  ?     Explain. 

6.  Are  x,  xy,  xy\  xf,  -■,  and  p'q-',  2A  PV,  Pg\  Q%  P~V  geo- 
metric series  ?     If  so,  name  a  and  r  in  each. 


314  RIQH  SCHOOL   ALGP.nnA  tCn.  XX 

7.  What  is  r  in  the  G.  P.  2,  |,  |,  •••  ?  in  the  G.  P.  21,  7,  J,  •••  ? 
Are  these  two  series  merely  parts  of  the  same  series  ?     Explain. 

8.  If  the  1st,  3d,  and  6th  terms  of  a  G.  P.  are  12,  3,  and  |, 
respectively,  find  r  and  then  write  down  the  first  8  terms  of  the 
series  (cf.  Ex.  26,  p.  311). 

201.  Formulas.  The  elements  of  a  G.  P.  are  connected  by 
the  two  fundamental  equations  shown  below  (cf.  §  198). 

Since  each  term  of  a  G.  P.  may  be  obtained  by  multiply- 
ing the  preceding  term  by  r  (cf.  Exs.  5  and  6,  p.  313), 
therefore,  if  I  represents  the  n{\\  term  of  such  a  series,  then 

l  =  ar-\  (1) 

Again,  if  s  represents  the  sum  of  a  G.  P.  of  n  terms,  then 
sz=a-{-  ar  -\-  ar^  -\-  ai^  +  •  •  •  +  «r"~^  +  ar^~^ ; 
whence,  multiplying  by  r,  we  obtain 

sr  =  ar  -^  ar^  +  a7^  -\- h  ar^~^  +  ar", 

and,  therefore,  by  subtracting  the  second  of  these  equations 
from  the  first,  member  from  member, 

s  —  sr  =  a—  ar" ; 

hence  ^~~/ *  (^) 

EXERCISE  CXXXV 

1.  By  means  of  §  201  (1)  write  down  the  6th  term  of  the  geo- 
metric series  4,  12,  36,  •  •  • . 

2.  As  in  Ex.  1,  write  the  7th  term  of  the  G.  P.  3,  6,  12,  .•• ; 
then  by  §  201  (2)  find  the  sum  of  the  first  7  terms  of  the  series. 
Check  results  by  writing  down  and  adding  these  first  7  terms. 

3.  Find  the  8th  term  of  24,  12,  6,  .••;  also  the  11th  term  of 
TTJinr?  ~  Tu^y   ttj"?  *"• 

Find  the  sum  of  the  following  series: 

4.  1,  2,  4,  .  •  •  ,  to  10  terms.  7.    1,  —  2  a;,  4  a^,  •  •  • ,  to  7  terms. 

5.  1,  1.5,  2.25,  •••,  to  6  terms.     8.    -5, -2, -.8,  •••,  to  A:  terms. 

6.  2,  —  I,   I,  ••• ,  to  7  terms.       9.   x,  x~^,  x~^j  •••  ,to  n  terms. 


200-201]  SERIES — THE  PIWGEESSIONS  315 

10.  Find  the  G.  P.  whose  3d  term  is  18,  and  whose  8th  term 
is  4374. 

Hint.  By  §  201  (1),  18  =  ar^,  and  4374  =  ar'' ;  hence,  dividing  the  second 
of  these  equations  by  the  first,  243  =  r^,  i.e.,  r  =  3. 

11.  Find  the  G.  P.  whose  oth  term  is  |-  and  whose  9th  term  is 
i||.     Also  find  the  sum  of  the  first  9  terms  of  this  series. 

12.  Show  that  §  201  (2)  may  be  written  in  each  of  the  follow- 
ing forms : 

g  (1  —  r")      a  —  rl      rl  —  a      ar"  —  a         -,   __a ar'' 

1  —  r    '     1  — r'     r— 1  '      r  — 1  '  1—r     1  —  r 

13.  By  actually  dividing  a(l  —  r")  by  1  —  r,  verify  the  correct- 
ness of  §  201  (2). 

14.  If  a  =  4,  I  —  972,  and  n  =  C,  find  r,  and  write  the  series. 

15.  If  71  =  12,  r  =  -  2,  and  6-  =  - 1365,  find  a  and  I 

16.  Find  the  sum  of  a  G.  P.  of  6  terms  whose  ratio  is  f  and 
whose  last  term  is  32. 

17.  If  r,  n,  and  I  are  given,  find  a  and  s ;  that  is,  find  a  and  s 
in  terms  of  r,  n,  and  I  (cf.  Ex.  16). 

18.  Find  a  and  I  in  terms  of  n,  r,  and  s;  also  r  and  s  in  terms 
of  a,  n,  and  /. 

19.  Is  a  G.  P.  fully  determined  when  any  three  of  its  elements 
are  given  (cf.  §  198,  Note)? 

20.  Three  numbers  whose  product  is  216  form  a  G.  P.,  and  the 
sum  of  their  squares  is  189.     What  are  the  numbers  ? 

Hint.     Let  -,  a,  and  ar  represent  the  required  numbers, 
r 

21.  Divide  38  into  three  parts  which  are  in  G.  P.,  and  such 
that  when  1,  2,  and  1  are  added  to  these  parts,  respectively,  the 
results  shall  be  in  A.  P. 

22.  Find  an  A.  P  whose  first  term  is  3,  and  such  that  its  2d, 
4th,  and  8th  terms  shall  be  in  G.  P. 

23.  If  the  population  of  the  United  States  was  76,000,000  in 
1900,  and  if  it  doubles  itself  every  25  years,  what  will  it  be  in 
the  year  2000? 

HIGH    SCH.    ALG.  — 21 


316  HIGH  SCHOOL  ALGEBRA  [Cn.  XX 

24.  Thinking  $1  per  bushel  too  high  a  price  to  pay  for  wheat, 
a  man  bought  10  bu.,  paying  3  cents  for  the  first  bushel,  6  cents 
for  the  second,  12  cents  for  the  third,  and  so  on.  What  did  the 
tenth  bushel  cost  him,  and  what  was  the  average  price  per 
bushel  ? 

25.  Show  that  the  amount  of  $  ^  for  n  years  at  a  given  rate 
{R)  of  compound  interest  is  the  nth  term  of  a  G.  P.  whose  first 
term  is  A  and  whose  ratio  is  (1  +  i?). 

26.  A  gentleman  loaned  a  friend  $250  at  the  beginning  of 
each  year  for  4  years.  If  money  is  worth  5  %  compound  interest, 
how  much  should  be  paid  back  to  him  at  the  end  of  the  fourth 
year  to  discharge  the  obligation  ? 

27.  The  president  of  a  charity  organization  starts  a  "  letter 
chain  "  by  writing  three  letters,  each  numbered  1,  requesting  each 
recipient  to  remit  10  cents  to  the  society,  and.  also  to  send  out 
3  other  letters,  each  numbered  2,  with  a  similar  request,  the  chain 
to  close  with  the  letters  numbered  20.  Should  every  recipient 
comply,  how  much  money  would  be  realized  ? 

202.  Infinite  decreasing  geometric  series.  A  decreasing 
G.  P.  is  one  in  which  r  <  1,  numerically,  and  in  which, 
therefore,  the  terms  grow  smaller  and  smaller  as  we  pass 
from  left  to  right  in  the  series.  Thus,  6,  3,  |,  f,  f,  •••  is  a 
decreasing  G.  P. 

If  we  let  s„  represent  the  sum  of  the  first  n  terms  of  this 
series,  then 


6     _  6(i)^  |-g^   j2,  p.  315 


l-\      1-1 

=  12-6(1)-^ 

and,  since  6(J)"~^  approaches  zero  as  a  limit  when  n  increases 
without  limit  [§  194  (i)],  therefore  s„  approaches  12  as  a 
limit  when  n  becomes  infinite ;    this  is  often  expressed  thus : 

8    =12. 


201-202]  SEBIES — THE  PROGBESSIONS  317 

Similarly,  whenever  r  <  1,    numerically,   and   7^  =  oo,    the 

formula 

_     a     _   gy" 

becomes  s    = ; 

*      1-r 

since  r"  approaches  the  limit  zero  as  n  becomes  infinite. 


EXERCISE  CXXXVI 

1.  If  in  Ex.  2,  p.  303,  AB  =  2  ft.,  show  that  the  successive 
distances   traversed,  when  expressed  in   inches,  form  the  G.  P. 

12fi.S3     3     3      3        3      ... 

±^,  D,  O,   2"?   4?    8^?  T6J    3T'         ' 

2.  By  §  201  (2),  find  s^  for  the  series  in  Ex.  1 ;  also  find  Sg,  Sg, 
^10,  and  s„. 

3.  From  Ex.  2,  p.  303,  show  that  in  the  series  of  Ex.  1  above 
s„  <  24,  no  matter  how  large  7i  may  be.  How  near  to  24  will  s„ 
approach  as  n  is  made  larger  and  larger  ?  Explain.  Also  find 
s^  by  §  202. 

4.  Find  s^  for  the  series  0.6,  0.06,  0.006,  •••,  and  thus  show 
that  0.6  {i.e.,  0.666  •••)  equals  f ;  similarly,  show  that  O.iS  (i.e., 
0.151515...)  equals  3%. 

Find  s^  for  each  of  the  following  series : 

5.  1,  -hh'--  9-   ^•^-  13.    l,k,k',^" 

6.  l,i,i,....  10.   0.12.  (wherein  A:  <1). 


7.   1,-1,  A,-.  11-   1.362.  14.   ^,-,-,, 


1    1^ 

8.    V2,  1,  V(X5,  ....        12.   4.7523.  (wherein  a; >  1). 

15.  If,  in  a  G.  P.,  r  is  positive  and  less  than  0.5,  show  that  any 
given  term  of  the  series  is  greater  than  the  sum  of  all  the  terms 
that  follow  it. 

16.  A  point  traversing  a  straight  line  moves  in  any  given 
second  75  %  as  far  as  in  the  preceding  second ;  if  it  moves  24  ft. 
in  the  first  second,  how  far  will  it  move  before  coming  to  rest  ? 


318  HIGH  SCHOOL  ALGEBRA  [Ch.  XX 

17.  If  a  sled  runs  80  ft.  during  the  first  second  after  reaching 
the  bottom  of  a  hill,  and  if  its  distance  decreases  20  %  during 
each  second  thereafter,  how  far  will  it  run  on  the  level  before 
coming  to  rest? 

18.  If  a  ball,  on  being  dropped  from  a  tower  window  100  ft. 
above  the  pavement,  rebounds  40  ft.,  then  falls  and  rebounds 
16  ft.,  and  so  on,  how  far  will  it  move  before  coming  to  rest  ? 

19.  Although  s^  for  the  series  ^,  i,  ^,  •••  is  1,  show  that  for  the 
series  -J,  J,  J,  ^,  •••,  6\  grows  larger  beyond  all  bounds,  by  suffi- 
ciently increasing  7i. 

Suggestion.    Write  the  series  thus :  s«  =  I  +(i  +  i)  +  (^  +  i  +  7  +  i)  -i , 

putting  8  terms  in  the  next  group,  16  in  the  next,  and  so  on,  and  show  that 
each  group  is  greater  than  |. 

203.  Geometric  means.    The  two  end  terms  of  a  finite  G.  P. 

are  called  its  extremes,  while  all  the  other  terms  are  called 

geometric  means  between  these  two. 

E.g.,  in  the  series  f,  i,  ^,  |,  and  ^,  the  extremes  are  |  and  ^4^,  and  |,  ^, 
and  I  are  geometric  means  between  them. 

In  a  G.  P.  of  three  terms,  the  (one)  geometric  mean  be- 
tween the  extremes  equals  the  square  root  of  their  product ; 
for  if  Gr  is  the  geometric  mean  between  a  and  5,  then 

—  =  -77,  [definition  of  a  G.  P. 

a       Gr 

whence  G-  =  ±  Vab, 

Ex.  1.  Find  the  geometric  mean  between  6  and  24;  also  between 
10  and  8. 


Solution.  The  geometric  mean  between  6  and  24  is  ±  V6  •  24, 
i.e.  ±12;  and  the  geometric  mean  between  10  and  8  is  ±  VlO  •  8, 
i.e.  ±4V5. 

Ex.  2.   Insert  four  geometric  means  between  f  and  —  -^. 

Solution.  In  this  series,  a  =  i  Z  =  —  y,  and  (since  four  means 
are  to  be  inserted)  n  =  4  +  2  =  6 ;  hence  by  §  201  (1),  -  -V-  =  |  •  ?-^, 
whence  ?*^  =  —  2^^-  and  r  =  —  |.  Therefore  the  required  series  is 
4    _2      1     _4.     9    __2 


3? 


1. -I.  I. -¥• 


202-204]  SERIES— THE  PROGRESSIONS  319 

EXERCISE  CXXXVII 

Find  the  geometric  mean  between  the  following  number-pairs : 

3.  18,8.  5.    },  -ff  7.    (a  +  b),(a-by, 

4.  5,20.  6.   0.5,3.5.  8.   2x-3,  {x-{-:iy. 
9.    Insert  4  geometric  means  between  3  and  96. 

10.  Insert  3  geometric  means  between  2  and  ^  (two  answers). 

11.  Insert  6  geometric  means  between  —  3125  x^^  and  ^^y  • 

12.  If  m  geometric  means  are  inserted  between  a  and  b,  show 
that  r  for  the  series  thus  formed  is  "''^b  -^  a. 

13.  What  does  the  formula  of  Ex.  12  become  when  m  =  l? 
Is  this  consistent  with  the  formula  for  G  obtained  in  §  203  ? 

14.  Two  numbers  differ  by  24,  and  their  arithmetical  mean 
exceeds  their  geometric  mean  by  6.     Find  the  numbers. 

204.  Harmonic  series.  An  harmonic  series,  or  harmonic 
progression  (H.  P.),  is  a  series  of  numbers  whose  reciprocals 
form  an  A.  P.  A  supposed  H.  P.  may  therefore  be  tested, 
and  problems  in  H.  P.  be  solved,  by  an  appeal  to  our 
knowledge  of  A.  P. 

E.g.,  the  numbers  f,  ^,  -^j,  ^'j,  •••  form  an  H.  P.  because  their  reciprocals, 
viz.,  I,  4,  J^,  -2/,  ...,  form  an  A.  P. 

Again,  if  we  were  asked  to  extend  the  H.  P.  f,  \,  y\,  f^,  •••  one  or  more 
terms  toward  the  right,  we  should  need  merely  to  form  the  corresponding 
A.  P.,  viz.,  I,  4,  -^5%  ■^,  ..-,  extend  it  as  required  (cf.  Ex.  3,  p.  308),  and 
then  write  the  reciprocals  of  its  terms. 

EXERCISE  CXXXVIII 

1.  If  the  6th  term  of  an  H.  P.  is  ^,  and  the  17th  term  is  ^, 

find  the  37th  term. 

Hint.  First  find  the  37th  term  in  an  A.  P.  whose  Oth  and  17th  terms 
are  3  and  ^,  respectively. 

2.  Insert  5  harmonic  means  between  2  and  —  3. 


320  HIGH  SCHOOL   ALGEBRA  [Ch.  XX 

3.  Assuming  x  to  be  the  harmonic  mean  between  a  and  6,  show 

that = ,  and  hence  that  x  =  — ^—  • 

X     a      b     X  a-^b 

4.  The  arithmetical  mean  between  two  numbers  is  5,  and  their 
harmonic  mean  is  3.2.     What  are  the  numbers? 

5.  The  difference  between  two  numbers  is  2,  and  their  arith- 
metical mean  exceeds  their  harmonic  mean  by  ^.  Find  the 
numbers. 

6.  Given  (b —  a):  (c  —  b)  =  a:  x,  prove  that  x  equals  a,  b,  or  c, 
according  as  a,  b,  and  c  form  an  A.  P.,  a  G.  P.,  or  an  H.  P. 

7.  If  a  and  b  are  two  unequal  positive  numbers,  and  ^  is  their 
arithmetical  mean,  G  their  geometric  mean,  and  .H  their  harmonic 
mean,  show  that :  (1)  A>  G>Hj  and  (2)  A:G=G:  H. 


CHAPTER   XXI 
MATHEMATICAL   INDUCTION  —  BINOMIAL   THEOREM 

205.  Proof  by  induction.  An  elegant  and  powerful  form 
of  proof,  and  one  that  is  very  useful  in  many  branches  of 
mathematics,  is  what  is  known  as  "proof  by  induction." 

To  illustrate :  suppose  it  to  have  been  found  by  trial  that 
a;  —  ^  is  a  factor  of  x^  —  ^^,  oi^  —  «/^  and  a;*  —  ^*,  and  that  we 
wish  to  know  whether  it  is  a  factor  of  x"  —  y^^  x^  —  y^^  ••• 
also.  Actual  trial  with  any  one  of  these,  say  ^  —  y^,  would 
show  that  it  is  exactly  divisible  hj  x  —  y,  but  besides  being 
somewhat  tedious,  this  division  gives  no  information  as  to 
whether  a;  — ^  is  or  is  not  a  factor  of  x^  —  y'^,  •••  also  ;  each 
successful  trial  increases  the  probability  of  the  success  of  the 
next,  but  it  proves  nothing  beyond  the  single  case  tried. 

That  X  —  y  is  a  factor  of  x^  —  ^™,  for  every  positive  inte- 
gral value  of  7i,  may  be  shown  as  follows  : 

Since  a^^  -  ^»  =  x(x''-^  -  y''-^)  +  y""'^  (x  -  y), 

therefore,  if  a;  — ^  is  a  factor  of  x^~^  —  y^~\  then  it  is  a  fac- 
tor of  the  second  member  of  this  equation,  and  therefore  of 
x^—y^  also  (why  ?)  ;  i.e.^  if  x  —  y  is  a  factor  of  the  differ- 
ence of  any  two  like  integral  powers  of  x  and  y,  then  it  is  a 
factor  of  the  difference  of  the  next  higher  powers  also. 

But  since,  by  actual  trial,  a;  —  ^  is  already  known  to  be  a 
factor  of  a;*  —  y^^  therefore,  by  what  has  just  been  proved,  it 
is  a  factor  of  a:^  —  ^^  also ;  again,  since  it  is  now  known  to  be 
a  factor  of  a^^  —  ^,  therefore  it  is  a  factor  oioc^  —  y^  -,  and  so 
on  without  end  :  ^.e.,  x  —  y  is  a  factor  of  ot^—y"^  for  every 
positive  integral  value  of  n, 

321 


322  BIGH  SCHOOL  ALGEBRA  [Ch.  XXI 

The  proof  just  given  is  an  example  of  what  is  known  as  a 
proof  by  mathematical  induction  ;  such  a  proof  consists  essen- 
tially of  two  steps,  viz. : 

(1)  Showing  hy  trial  or  otherwise  the  correctness  of  a  given 
law  when  applied  to  one  or  more  particular  cases^  and 

(2)  Proving  that  if  this  law  is  true  for  any  given  case,  then 
it  is  true  for  the  next  higher  case  also. 

From  (1)  and  (2)  it  then  follows  that  the  proposition 
under  consideration  is  true  for  all  like  cases.* 

EXERCISE  CXXXIX 

1.  Prove  that  the  sum  of  the  first  n  odd  integers  is  n^. 

Solution.     (1)  By  trial  it  is  found  that  1  +  3  =  2^  and  1  +  3  +  5  =  32. 

(2)  Moreover,  if  i  +  3  +  5  +  . . .  +  (2  A:  -  1)  =  A;^, 

then,  by  adding  the  next  odd  integer  to  each  member,  we  obtain 

1+  3  +  5  ...  +  (2  A:  -  1)  +  (2  A;  +  1)  =:  ^•2  +  (2  A:  +  1)  =  (A;  +  1)2 ; 
i.e.,  if  the  law  in  question  is  true  for  the  first  k  odd  integers,  then  it  is  true 
for  the  first  k  +  I  odd  integers  also. 

But,  by  actual  trial,  this  law  is  known  to  be  true  for  the  first  3  odd  inte- 
gers, hence  it  is  true  for  the  first  4  ;  and,  since  it  is  now  known  to  be  true 
for  the  first  4,  therefore  it  is  true  for  the  first  6  ;  and  so  on  without  end  : 
hence  the  sum  of  any  number  of  consecutive  odd  integers,  beginning  with  1, 
equals  the  square  of  their  number. 

By  mathematical  induction  prove  that : 

2.  14-2  +  3+.. -+71  =  1.71(71  +  1). 

3.  2  +  4  +  6+.-.  +  27i  =  7i(7i+l). 

4.  12  +  2^  +  32+.. .  +  7l2  =  ^n(71  +  l)(271  +  l). 

*  The  student  should  carefully  distinguish  between  mathematical  induc- 
tion, as  here  defined,  and  what  is  known  as  inductive  reasoning  in  the 
natural  sciences.  A  proof  by  mathematical  induction  is,  from  its  very  nature, 
absolutely  conclusive.  On  the  other  hand,  the  inductive  method  in  physics, 
chemistry,  etc.,  consists  in  formulating  a  statement  of  a  law  which  will  fit 
the  particular  cases  that  are  known,  and  regarding  it  as  a  laio  only  so  long 
as  it  is  not  contradicted  by  other  facts  not  previously  taken  into  account. 
From  the  nature  of  the  case  step  (2)  above  cannot  be  applied  in  physics,  etc. 


205-206]  MATHEMATICAL  INDUCTION  323 

5.  l«  +  2«  +  3«+"-+^'  =  i-n2(n  +  l)2  =  (l+2  +  3+--+ri)'. 

6.  A  +  A4-A+..-+         ' 


1.2     2.3      3-4  n(n-\-l)      n  +  1 

7.  1.2  4-2.3  +  3.4+...+w(n  +  l)  =  in(n  +  l)(w.-f-2). 

8.  a  +  ar  +  ar^-] f- ar^'-i  =  ^^i^— 1^ . 

1  —  r 

9.  a;"  —  ?/''  is  divisible  by  ic  +  2/  when  71  is  even. 

10.  Having  established  (1)  and  (2)  in  the  inductive  proof  of 
any  law,  show  the  generality  of  the  law  by  showing  that  there 
can  be  no  Jirst  exception,  and  therefore  no  exception  whatever. 

206.  The  binomial  theorem.  The  method  of  induction 
furnishes  a  convenient  proof  of  what  is  known  as  the  bino- 
mial theorem;  this  theorem,  which  was  presented  without 
formal  proof  in  §  112,  may  be  symbolically  stated  thus : 

wherein  x-\-y  represents  any  binomial  whatever,  and  n  is 
any  positive  integer. 

To  prove  this  theorem  by  mathematical  induction,  observe 
first  that  it  is  correct  when  n  =  %  for  it  then  becomes 

2  ^  •  1 

(x  +  ?/)2  =  a;2  _|_  ^^  _^  rL_^ a;V ;  i'e.,{x  -^  yy^  =  x^  -\-1xy  -^  ?/2, 

which  agrees  with  the  result  of  actual  multiplication. 

Again,  if  (1)  is  true  for  any  particular  value  of  n^  say  for 
n  =  k^  i.e.,  if 

{X  -f  yy  =  x^-\-\  x'-^y  +  ^^f=^  x^'Y 

+^^^-y-^> .-¥+-,    (2) 


324  HIGH  SCHOOL  ALGEBRA  [Ch.  XXI 

then,  on  multiplying  each  member  of  (2)  by  x-\-y^  it  be- 
comes 

{X + yy^^ = x'+^  +  J  x'y  +  ^^^"^^  x'-y 

1.2-3  "^ 

which  is  of  precisely  the  same  form  as  (2),  merely  having 
k-\-l  wherever  (2)  has  k.  Moreover,  (3)  is  obtained  from 
(2)  by  actual  multiplication,  and  is  therefore  true  if  (2)  is 
true ;  hence,  if  the  theorem  is  true  when  the  exponent  has  any 
particular  value  (sa}^  A;),  then  it  is  also  true  when  the  exponent 
has  the  next  higher  value. 

But,  by  actual  multiplication,  the  theorem  is  known  to  be 
true  when  7i=  2,  hence,  by  what  has  just  been  proved,  it  is 
true  when  ?i  =  3 ;  again,  since  it  is  now  known  to  be  true 
when  n  =  3,  therefore  it  is  true  when  n  =  4  ;  and  so  on  with- 
out end :  hence  the  theorem  is  true  for  every  positive  inte- 
gral exponent,  which  was  to  be  proved. 

EXERCISE  CXL 

1.  In  the  expansion  of  (x  -f-  ?/)",  what  is  the  exponent  of  y  in 
the  2d  term?  in  the  3d  term?  in  the  4th  term?  in  the  i2th 
term  ?  in  the  rth  term  ?  What  is  the  sum  of  the  exponents  of 
X  and  y  in  each  term? 


20r,-207]  BINOMIAL    THEOREM  325 

2.  In  the  expansion  of  (x  +  ?/)**  what  is  the  largest  factor  in 
the  denominator  of  the  3d  term  ?  of  the  4th  term  ?  of  the  10th 
term  ?  of  the  rth  term  ?  In  any  given  term,  how  does  this  factor 
compare  with  the  exponent  of  y? 

3.  In  the  expansion  of  (aj-f?/)'*,  what  is  subtracted  from  n  in 
the  last  factor  of  the  numerator  in  the  3d  term  ?  in  the  4th  term  ? 
in  the  5th  term  ?  in  the  9th  term  ?  in  the  rth  term  ? 

4.  Based  upon  your  answers  to  Exs.  1-3,  write  down  the  6th 
term  of  (x  +  yy.  Also  write  the  10th  term ;  the  17th  term ;  and 
the  ?-th  term. 

207.  Binomial  theorem  continued.  Strictly  speaking,  all 
that  was  really  proved  in  §  200  is  that,  for  every  positive 
integral  value  of  the  exponent,  the  first  four  terms  of  the 
expansion  follow  the  law  expressed  by  (1)  ;  that  all  the 
terms  follow  this  law  will  now  be  shown. 

In  multiplying  (2)  of  §  206  by  a;  +  ?^,  the  2d  term  of  the 
product  (3)  is  x  times  the  2d  term  plus  i/  times  the  1st  term 
of  (2) ;  so,  too,  the  10th  term  of  (3)  would  be  found  by 
adding  x  times  the  10th  term  to  ?/  times  the  9th  term  of  (2), 
and  the  rth  term  of  (3)  by  adding  x  times  the  rth  term  to 
1/  times  the  (r  —  l)th  term  of  (2). 

But  the  (r  — l)th  and  the  rth  terms  of  (2)  are,  respec- 
tively, 

1.2.3....(r-2)  ^ 

and        K^-l)(^-2)...(^-r+3)(y^-r  +  2)    ,_,^,      , . 
1.2.3.  •(r-2Xr-l)  ^      ' 

therefore  the  rth  term  of  (3)  is 

'k(k  -  l)(Jc  -  2)  ...  (k  -  r  -\-  ^) 
1.2.3.  ...(r-2) 

^(^-l)(^-2)...(^-r  +  3)(^-r  +  2)1    ,_,^,      ^ 
1.2.3. ...(r-2)(r-l)  J  ^     ' 

(k+l}k(k-l)  ...  (^-r+  3)   ,_,^,      1 
1.2.3. ...(r-1)  "^        ^     ' 


326  IIIGII   SCHOOL   ALGEBRA  [Cn.  XXI 

wliicli  conforms  to  the  law  for  the  rth  term  expressed  by  (1) 
of  §  206.  Hence  the  rth  term,  i.e.^  every  term,  in  (3)  con- 
forms to  the  law  expressed  by  (1),  which  was  to  be  proved. 

EXERCISE  CXLI 

1.  Write  down  the  expansion  of  (a+by-,  also  of  {p  —  qf. 
Explain  why  the  alternate  terms  in  the  expansion  of  (^  —  qf  are 
negative. 

2.  Write  down  the  1st,  2d,  3d,  and  8th  terms  of  {x  +  yf^. 

3.  Write  down  the  4th  and  7th  terms  of  {a  —  xy\ 

4.  How  many  terms  are  there  in  the  expansion  of  (x+yy^? 
Write  down  the  first  three,  and  also  the  last  three  terms  of  this 
expansion,  and  compare  their  coefficients. 

5.  Write  down  the  coefficient  of  the  term  containing  ay,  in 
the  expansion  of  (a  —  yy^. 

6.  Expand  (3  a^  -  2  xff  ;  compare  Ex.  2,  §  57. 

7.  Write  down  the  4th  and  9th  terms  of  (f  ic  —  |  ?/)". 

8.  How  many  terms  are  there  in  [a; )    ?     Write  down  the 

\       xj  _ 

10th  term.     Also  write  down  the  5th  term  of  (J^-^J'^J. 

9.  Write  down  the  term  of  (3  a^-^-  2  x'y,  i.e.,  of  (xy(S  a^-2)^ 
which  contains  a^^. 

10.   Write  down  the  term  of  f  a^ )    which  contains  a". 


3  a, 

11.  Expand  (a^  +  3  a^x'^y,  and  write  the  result  with  positive 
exponents. 

12.  Expand  (l  —  x-{-  of)*  by  means  of  the  binomial  theorem 
(cf.  Exs.  40-41,  p.  176). 

13.  By  applying  the  law  expressed  in  (1)  of  §  206  show  that 
the  coefficient  of  the  (n  +  l)th  term  of  (x  -f  yy  is  1 ;  also  show 
that  the  coefficient  of  every  term  thereafter  contains  a  zero  factor, 
and  hence  that  {x+yy  contains  only  n-\-l  terms. 


207-208]  MATHEMATICAL  INDUCTION^   ETC,  327 

14.  Show  that  the  sum  of  the  binomial  coefficients,  ^.e.,  of  1, 
71    n(n-l)    n(n-l)(n-2)  ■    o,, 

r     2~~'     1.2.3     '••'^'^- 

Hint.     After  expanding  (a;  -f  y)**,  let  cc  =  y  =  1. 

15.  Show  that  the  sum  of  the  even  coefficients  (i.e.,  the  2d, 
4th,  . .  •)  in  Ex.  14  equals  the  sum  of  the  odd  coefficients,  and  that 
each  sum  is  2"~\ 

Hint.     In  (x  +  2/)**  let  x  =  1  and  y  =—  1. 

16.  Show  that  the  coefficient  of  the  rth  term  in  (x  4-?/)"  may  be 

obtained  by  multiplying  that  of  the  (r— l)th  term  by  ^~^"'~    , 

?'  — 1 
and  thus  show  that  the  binomial  coefficients  increase  numerically 
in  going  from  term  to  term  toward  the  center. 

17.  Show  that  the  coefficient  of  the  rth  term  is  numerically 
greater  than  that  of  the  (r  —  l)th  term  so  long  as  r  <  ^  (n  4-  3)  ; 
and  thus  write  down  the  term  whose  coefficient  is  greatest  in  the 
expansion  of  (x  +  3/)" ;  and  also  in  (x  +  ijy^. 

208.  Binomial  theorem  extended.  It  may  be  remarked  in 
passing  that  the  binomial  theorem  (§  206),  which  has  thus 
far  been  restricted  to  the  case  where  the  exponent  is  a  posi- 
tive integer,  is  greatly  extended  in  Higher  Algebra,  where 
it  is  shown  that  under  certain  restrictions  it  admits  negative 
and  fractional  exponents  also.  Although  the  proof  of  this 
fact  is  beyond  the  limits  of  this  book,  its  correctness  may  be 
assumed  in  the  following  exercises. 

EXERCISE  CXLII 
Using  the  binomial  theorem,  write  the  first  5  terms  of : 

1.  (a;4-2/)i  3.    (a-c)i  5.    {l-as^^. 

2.  (l  +  w)i  4.    (a-^b)-\  6.    (2m-Jc)-K 

7.  Write  the  6th  term  of  (3  r  —  sy^ ;    also  the  5th  term  of 
{\-3x)\ 

8.  Show  that  in  such  cases  as  the  above  the  binomial  theorem 
leads  to  infinite  series  (cf.  Ex.  13,  p.  326).  , 


328  HIGH  SCHOOL  ALGEBRA  [Ch.  XXI 

9.  Expand  (1  —  x)~'^  to  8  terms  by  the  binomial  theorem  and 
compare  the  result  with  the  first  8  terms  of  the  quotient 
1^(1-0;). 

10.  Show  that,  when  expanded  by  the  binomial  theorem  and 

simplified,  (25  + 1)^  =  5  +  ^^  -  wo^  +  sirio^ >  compare  this 

result  with  VW  as  found  by  the  usual  method. 

11.  By  expanding  (9  —  2)^,  find  an  approximate  value  of  V7; 
similarly,  find  an  approximate  value  of  VM  (i.e.,  V27  +  4),  and 
of  ^40  (i.e.,  ^^2  +  8). 

209.  The  square  of  a  polynomial.  In  §  bQ  it  was  pointed 
out  that,  by  actual  multiplication,  the  square  of  a  polyno- 
mial consisting  of  3,  4,  or  5  terms  equals  the  sum  of  the 
squares  of  all  the  terms  of  the  polynomial,  plus  twice  the 
product  of  each  term  by  all  those  that  follow  it.  It  will 
now  be  shown  that  if  this  theorem  is  true  for  polynomials  of 
n  terms,  then  it  is  also  true  for  those  of  w  +  1  terms ;  and 
from  this  it  will  follow,  as  in  §  205,  that  it  is  true  for  poly- 
nomials of  any  finite  number  of  terms  whatever,  since  it  is 
already  known  to  be  true  for  polynomials  of  five  terms. 

Let  a+h-[-  c-\-  -•■  -\-  p -[-  qhQ  2i  polynomial  of  n  terms,  and 

let  (a+h-\-c-\-  •••  +Jt?4-g)2=  «2  +  h^-\ \- q^ -\- 2  ah -{■  2  ac+'-' 

-\-1aq-\-1hc-\ h  2^g  + [-2jt?g. 

In  this  identity  replace  a  everywhere  by  ic  +  ^  ;  then  the 
number  of  terms  in  the  polynomial  in  the  first  member  will 
become  n  +  1,  and  the  second  member  will  still  consist  of 
the  sum  of  the  squares  of  all  the  terms  of  the  polynomial, 
plus  twice  the  product  of  each  term  by  all  those  that  follow 
it  (the  student  should  work  this  out  in  detail)  ;  therefore, 
if  the  theorem  is  true  for  polynomials  of  n  terms,  then  it  is 
also  true  for  those  of  n  +  1  terms,  which  was  to  be  proved. 


CHAPTER    XXII 
LOGARITHMS 

210.  Introduction.  Early  in  tlie  seventeenth  century,  two 
British  mathematicians,  Lord  Napier  and  Henry  Briggs,  con- 
ceived the  idea  of  expressing  all  real  positive  numbers  as 
powers  of  10,*  arranging  the  exponents  of  these  powers  in  a 
table  for  convenient  reference,  and  then  employing  this  table 
to  simplify  certain  arithmetical  computations,  especially 
multiplication. 

E.g.,  to  find  the  product  of  3.578,  7.986,  and  48.67,  we  find 
from  the  table  that 

3.578  =  10°^^,  7.986  =  10«-^23^  and  48.67  =  lO^^^^^ 
whence    3.578  x  7.986  x  48.67  =  10«^^  x  lO"^^^  x  lO^-^^ 

__  -j^QO.5536+0.9023+1.6873  Tfi   3Q 

we  now  find  from  the  table  that 

10«-i^  =  1390.6, 
whence    3.578  x  7.986  x  48.67  =  1390.6. 

Thus,  by  performing  an  addition  (of  the  exponents),  we  have 
found  the  product  of  the  given  numbers. 

Other  advantages  of  such  a  table  of  exponents  (loga- 
rithms) will  be  shown  later  (§  218) ;  some  necessary  defini- 
tions and  principles  must  now  be  given. 

211.  Definitions.  The  logarithm  of  a  number  (iV)  to  any 
given  base  (5)  is  the  exponent  (x)  of  the  power  to  which 
this  base  must  be  raised  to  equal  the  given  number. 

*  That  it  is  possible  to  do  this,  either  exactly  or  to  any  required  degree  of 
approximation,  will  be  assumed  in  this  chapter. 

329 


330  HIGH  SCHOOL  ALGEBRA  [Ch.  XXII 

The   logarithm    of   iV  to   the   base   b   is   usually   written 
logjiV;   and  the  two  statements 

iV=5^    and    log(,N=x 
are,  therefore,  only  different  ways  of  saying  the  same  thing. 


E.g.,      •.•2^  =  8,       .•.log28  =  3;       •.•3^  =  243,      .•.log8243  =  5 
and  •.•  10i-^'3  =  48.67,     .-.  logjo  48.67  =  1.6873. 


EXERCISE  CXLIII 

1.  rrom  the  equation  3^*  =  81,  find  logg  81 . 
Translate  into  logarithmic  equations  (cf.  Ex.  1) : 

2.  4^=64.  5.  2^  =  32.  8.   10«  =  1. 

3.  9^  =  81.  6.   (|)'  =  ^V  9-   10-' =  .001. 

4.  10-*  =  1000.  7.  2-^  =  ^2-  10-   (i)"'  =  125. 
Express  the  following  statements  in  the   exponent   notation, 

and  then  verify  the  correctness  of  each : 

11.  log7  49  =  2.  14.   Iogiol0  =  l.  17.  log3i  =  -2. 

12.  log2l6  =  4.  15.  logiol  =  0.  18.  logio.0001  =  -4. 

13.  log.5 .125  =  3.       16.  Iogiol0000  =  4.     19.  loga256  =  -8. 

20.  Find  the  value  of  the  following  logarithms :  logg  27 ; 
log2  64;      log_8  64;      log_6(-216);      log4l;      logio.l;      log.ilO; 

21.  Between  what  two  consecutive  integers  does  each  of  the 
following  logarithms  lie:  logio83;  logio2224;  logio4;  logio.007; 
logio  .1256  ?     Explain  your  answers. 

22.  May  the  base  of  a  set  of  logarithms  be  fractional  ?  nega- 
tive ?  May  a  logarithm  itself  be  fractional  ?  negative  ?  May 
negative  numbers  have  logarithms  ?     Illustrate  your  answers. 

212.  Principles  of  logarithms.  Since  logarithms  are  expo- 
nents (§  211),  therefore  the  principles  of  logarithms  are 
easily  obtained  from  those  governing  exponents  (§§  171-175). 


211-212]  LOGARITHMS  331 

Principle  1.     TJie  logarithm  of  1  to  any  base  is  0,  and  the 
logarithm  of  the  base  itself  is  1 ;  i.e.^ 

logj  1  =  0  and  log^  5  =  1. 
The  correctness  of  tliis  principle  follows  at  once  from  the 
definition  of  a  logarithm  (§  211),  and  from  the  fact  that 

Z>o=l  and  51  =  6.  [§§  173,  9 

Principle  2.      The  logarithm  of  a  product  equals  the  sum 
of  the  logarithms  of  the  factors  ;  i.e.^ 

log,  (M/if)  =log, M -{-log, /if. 

For,  if  M=  b"^  and  iV^=  ^>^ 

then  M]sr=  b^'^\  [•/  b''  -  by  =  b'^+y 

whence  log<.  {MN)  =x-\-y  =  log^  M-\-  log^  N. 

Similarly,  log (MZVP  •  •  •)  =  log^  M+  log^, iY+  log, P  +  •  •  •• 

Let  the  pupil  translate  Principles  3-5  below  into  verbal 
language,  and  prove  each  in  detail  (cf.  Principle  2  above). 

Principle  3.     log^^=log;,  Af-log^/l^. 

Hint.    11  M=h''  and  N  =  &",  then  M^  N=  b''-^. 

Principle  4.     log^  N^  =  p'  log^  N. 

Hint.     liN=  &^,  then  N^  =  (b^y  =  b^». 

Principle  5.    log^  ^N  =  ~  •  logj  N. 

r 

1  _        1 

Hint.     If  iV^=  ^)^  then  Vi\^=(^,-)^  1</N=N' 


EXERCISE  CXLIV 

Using  Principles  1-5,  express  the  following  logarithms  in 
terms  of  log  a,  log  c,  and  log  e,  the  base  h  being  understood 
throughout : 

1.  log(ac).  3.    log  (ace^).  ^  ^^ 

2.  log(a^).  4.   log  (cV).  *  c* 

HIGH   8CH.    ALG.  —  22 


332  HIGH  SCHOOL  ALGEBRA  [Ch.  XXII 

6.    log-.  alogVc.  11.   log(aV^). 


a 


6/ lo       1 3  6 


9.    log  ^ace.  12.   log 

lo    i.  12  ^"^ 


a  10.    log  (a^c^).  13.    logVce~l 

Express  each  of  the  following  by  means  of  a  single  logarithm, 
and  explain  [e.g.,  log  c  +  log  e  =  log  (ce)]  : 

14.  logc  +  loge.  16.   2  log  a +  3  log  e.    is.   i(loge  —  log  5  a). 

15.  log  c  — log  e.  17.    4(logc  —  loga).     19.   f  log  a -f  4  log  2  c. 

If  logio  2  =  0.3010,  logio  3  =  0.4771,  and  logi„  7  =  0.8451,  find  the 
logarithms,  to  the  base  10,  of  : 


20. 

6(i.e., 

3. 

2). 

25. 

63. 

30. 

400. 

21. 

14. 

26. 

f. 

31. 

tV 

22. 

42. 

27. 

2i. 

32. 

V3. 

23. 

49. 

28. 

5(/.e.,  i^y 

33. 

<m. 

24. 

12(i.e. 

,2^ 

'.3). 

29. 

30(?.e.,  3 . 

10). 

34. 

5-2^. 

•  (i)^. 

213.  Common  logarithms ;  characteristic  and  mantissa. 
Logarithms  to  the  base  10  (called  common  or  Briggs  loga- 
rithms) possess  many  advantages  over  those  having  any  other 
base,  and  are  used  in  all  practical  computations.  In  the 
following  pages  the  base  10  will  be  understood  when  no  base 
is  written  ;  thus  log  25  will  mean  log^^  25. 

Now  since  100  =  1,  10^  =  10,  102  =  100,  10^=1000,  etc., 
therefore  log  1  =  0,  log  10  =  1,  log  100  =  2,  log  1000  =  3, 
etc.;  and  therefore  the  logarithm  of  any  number  between  1 
and  10  lies  between  0  and  1,  i.e.,  it  is  0  plus  a  decimal ;  the 
logarithm  of  any  number  between  10  and  100  is  1  plus  a 
decimal ;  the  logarithm  of  any  number  between  100  and 
1000  is  2  plus  a  decimal ;  etc. 

Again,  since  10-i=.l,  10-2=. 01,  10-3  =.001,  etc., 
therefore  log.l  =  -l,  log  .01  =  -2,  log  .001  = -3,  etc., 
and  therefore  the  logarithm  of  any  number  between  1  and  .1 
is  —  1  plus  a  decimal ;  the  logarithm  of  any  number  between 
.1  and  .01  is  —  2  plus  a  decimal  ;  etc. 


212-214]  LOGARirUMS  333 

The  integral  part  (whether  positive  or  negative)  of  the 
logarithm  of  a  number  is  called  the  characteristic  of  the 
logarithm,  and  the  decimal  part  (always  positive)  is  called 
the  mantissa. 

E.g.^  log  685  is  2.8357  ;  the  characteristic  is  2,  and  the  mantissa  is  .8357. 

214.  Advantages  of  the  base  10.  (i)  It  follows  from  §  213 
that  the  characteristic  of  the  logarithm  of  any  number 
between  10  and  100  is  1  (why  ?);  between  100  and  1000,  2 ; 
between  10,000  and  100,000,  4;  between  .01  and  .001,  -3 
(why?);  between  .001  and  .0001,  —4;  etc.;  i.e.  (let  pupil 
fully  explain  why), 

( 1)  The  characteristic  of  the  logarithm  of  any  number  greater 
than  unity  is  less  by  one  than  the  number  of  digits  in  its  integral 
part ;  and  (2)  the  characteristic  of  the  logarithm  of  any  number 
less  than  unity  is  negative,  and  (^numerically}  greater  by  one 
than  the  number  of  ciphers  preceding  the  first  significant  figure 
of  the  given  number. 

(ii)    Another  great  advantage  of  logarithms  to  the  base 

10  is  that  moving  the  decimal  point  to  the  right  or  left  in 

any  given  number  makes  no  change  in  the  mantissa  of  the 

logarithm  of  that  number. 

^.^.,  if  log  57.32  =1.7583, 

then  log  573.2  =  2.7583 ;.  r§  212, 

for  log  573.2  =  log  (57.32  x  10)=  log  57.32  +  log  10     LPrin.  2 

=  1.7583  +  1=2.7583. 

EXERCISE  CXLV 

1.  Which  of  the  following  logarithms  have  negative  charac- 
teristics:  log  79;  log  .315;  log  5228;  log  4.07;  log  .00098  ; 
log  .0231 ;  log  14865.01  ?     Explain. 

2.  How  many  units  in  the  characteristic  of  each  of  the  above 
logarithms  ?     Explain. 

3.  How  many  places  to  the  left  of  the  decimal  point  has  a 
number  if  the  characteristic  of  its  logarithm  is  2?  5?  0?  7? 
Explain  your  answers. 


334  HIGH  SCHOOL  algebra  [Ch.  XXII 

4.  How  many  ciphers  has  a  decimal  to  the  left  of  its  first 
significant  figure  if  the  characteristic  of  its  logarithm  is  —  5  ? 
—  1  ?    —  3  ?     Explain  your  answers. 

5.  If  log  469  =  2.6712,  show  that  log  469000  =  5.6712  and  that 
log  4.69  =  0.6712  [cf.  §  214  (ii)].  Also  point  out  the  characteristic 
and  the  mantissa  in  each  of  these  logarithms. 

6.  If  log  8.93  =  0.9509,  find  log  .0893;  log  893000;  log  89.3; 
and  log  .000893. 

7.  Show  that  logarithms  to  the  base  10  have  at  least  two  prac- 
tical advantages  over  logarithms  to  other  bases. 

215.  Table  of  common  logarithms.  The  mantissas  (with 
decimal  points  omitted)  of  the  logarithms  of  all  integers  be- 
tween 1  and  1000  are  given  in  tabular  form  on  pp.  341,  342. 

This  table  omits  the  characteristics  of  the  logarithms  be- 
cause these  can  be  supplied  by  inspection  (§  214)  ;  but  it 
includes  the  mantissas  of  the  logarithms  of  decimal  fractions 
as  well  as  of  integers — the  mantissa  of  log  26.5,  or  of  log 
.00265,  for  example,  is  the  same  as  the  mantissa  of  log  265. 

Note.  Except  where  a  number  is  an  integral  power  of  10,  the  mantissa 
of  its  logarithm  is  an  endless  decimal.  Hence  the  mantissas  in  a  table  are 
only  approximate  values,  correct  to  three  or  more  decimal  places.  The  table 
on  pp.  341,  342  is  called  a  "four-place  table  "  because  the  mantissas  are  com- 
puted to  four  decimal  places.  Such  a  table  gives  results  less  accurate  than 
those  obtained  from  a  six-place  table,  for  example ;  the  degree  of  accuracy 
required  in  any  given  computation  determines  the  choice  of  table. 

216.  Use  of  tables.  Given  a  number,  to  find  its  logarithm. 
Write  down  the  characteristic  by  §  214,  before  consulting  the 
table.     Then : 

(a)  The  number  consisting  of  not  more  than  three  significant 
figures.  Find  the  first  two  figures  of  the  number  in  the 
column  headed  N  in  the  table  ;  opposite  these,  and  in  the 
column  headed  by  the  third  figure  of  the  given  number,  find 
the  required  mantissa. 


214-210]  LOGARITHMS  335 

Ex.  1.   Find  log  374. 

Solution.  By  §  214  the  characteristic  is  2.  On  p.  341  oppo- 
site 37  (in  the  N  column),  and  in  the  column  under  4,  we  find  the 
mantissa  .5729 ;  hence  log  374  is  2.5729. 

Ex.  2.   Find  log  .835. 

Solution.  The  characteristic  is  —  1  (§  214).  Now  the  man- 
tissa of  log  .835  is  the  same  as  the  mantissa  of  log  835  (§  214), 
which,  found  as  in  Ex.  1,  is  .9217 ;  hence  log  .835  is  —  1  +  .9217. 

Note.  The  mantissa  being  always  positive,  the  sign  of  a  negative  charac- 
teristic is  (to  prevent  confusion)  written  over  the  characteristic.  Thus  we 
write  log  .835  not  as  -  1  +.9217,  but  as  1.9217. 

Another  common  notation  is  to  add  10  to  the  negative  characteristic  and 
then  to  indicate  the  subtraction  of  10  from  the  entire  logarithm.  Thus 
log  .836  may  be  written  9.9217  -  10. 

Ex.  3.    Find  log  6. 

Solution.  The  characteristic  is  0  (§  214)-  The  mantissa  of 
log  6  is  the  same  as  that  of  log  600,  which  is  .7782 ;  hence  log  6 
is  0.7782. 

(5)  The  number  consisting  of  more  than  three  significant 
figures.  In  this  case  we  assume  that  the  logarithm  of  a 
number  varies  directly  as  the  number  itself.  While  this  as- 
sumption is  not  entirely  correct  (doubling  50,  for  example, 
multiplies  its  logarithm  by  1.17  H-  instead  of  by  2),  still  for 
small  changes  in  a  number,  it  leads  to  results  sufficiently 
accurate  for  many  purposes. 

Ex.4.   Find  log  2547. 

Solution.  The  characteristic  is  3,  and  the  mantissa  is  the 
same  as  that  of  log  254.7  (§  214). 

Now,  mantissa  of  log  254  is  .4048, 

and  mantissa  of  log  255  is  .4065, 

i.e.,  adding  1  to  254  adds  .0017  to  the  mantissa  of  its  logarithm, 
hence  adding  .7  to  254  should  add,  approximately,  .7  of  .0017,  i.e., 
.0012,  to  its  logarithm,  and  hence 

log  2547  =  3.4048  +  .0012  =  3.4060. 


836  IIIGTI  SCHOOL  ALGEBIIA 

Ex.  5.    Find  loff  74.326. 


[Ch.  XXII 


SoLUTio:^.  The  characteristic  is  1,  and  the  mantissa  equals 
the  mantissa  of  log  743.26,  which  equals  (let  pupil  explain  why) 
mantissa  of  log  743  -f  .26  X  (log  744  -log  743). 

=  .8710  +  . 26  X. 0006, 

=  .8712; 
hence  log  74.326  =  1.8712. 


EXERCISE 

CXLVI 

By  reference  to  the  table 

verify 

that : 

6.   log  416 

=  2.6191. 

9. 

log 

.00972  =  3.9877. 

7.   log  5 

=  0.6990. 

10. 

log 

5268     =3.7216. 

8.   log  83000 

=  4.9191. 

11. 

log 

.7436  =1.8714. 

Find  the  logarithm  of  : 

12.   513. 

19. 

7. 

26.   .1008. 

13.   692. 

20. 

.009. 

27.   3.141. 

14.   3.47. 

21. 

4000. 

28.   22220. 

15.   .81. 

22. 

36.02. 

29.    .000694. 

16.   27.8. 

23. 

6215. 

30.    .011111. 

17.   .055. 

24. 

.3972. 

31.   437910. 

18.   200. 

25. 

851.3. 

32.   .0018952. 

33.  Write  the  logarithms  in  Exs.  24,  2&,  29,  30,  and  32  in 
two  different  forms  (cf.  Ex.  2,  Note). 

217.  Given  a  logarithm,  to  find  the  corresponding  number. 
The  number  to  which  a  given  logarithm  corresponds  is 
called  its  antilogarithm.     Thus, 

•.  •  log  53  =  1.7243,     .  •.  antilog  1.7248  =  53. 

Antilogarithms  are  found  by  reversing  the  processes  of 
§  216  ;  a  few  examples  will  make  the  procedure  plain. 


216-217]  LOGARITHMS  337 

Ex.  1.   Find  antilog  2.5587. 

Solution.  On  consulting  the  table  we  find  that  .5587  is  the 
mantissa  of  log  362,  and  the  characteristic  2  tells  us  that  there 
must  be  one  cipher  between  the  decimal  point  and  the  first  sig- 
nificant figure  [§  214  (ii)]  ;  hence 

antilog  2.5587  =  .0362. 

Ex.  2.   Find  antilog  1.7493. 

Solution.     On  consulting  the  table  we  find  that 
.7490  =  mantissa  of  log  561, 
and  .7497  =  mantissa  of  log  562, 

these  being  the  mantissas  next  smaller  and  next  larger,  respec- 
tively, than  the  given  mantissa.  Hence  antilog  1.7493  lies  be- 
tween 56.1  and  56.2  (the  characteristic  being  1). 

Again,  since  the  given  mantissa,  viz.,  .7493,  is  f  of  the  way 
from   .7490  to   .7497,  therefore    the    required    antilogarithm    is 
approximately  f  of  the  way  from  56.1  to  56.2, 
i.e,,  antilog  1.7493  =  56.1  +  f  of  0.1 

=  56.1 -f  .043 
=  56.143. 

Ex.  3.   Find  antilog  3.1188. 
Solution.  antilog  3.1206  =  1320, 

antilog  3.1173  =  1310 
whence,  subtracting,  we  obtain  .0033  and  10 
also  3.1188  -  3.1173  =  .0015 ; 

therefore  antilog  3.1188  =  1310  +  ^f  of  10 

=  1310  4-  4.5  =  1314.5. 

EXERCISE  CXLVII 

Verify  from  the  table  that : 

4.  antilog  0.1875  =  1.54.  6.    antilog  1.8454  =  70.05. 

5.  antilog  1.6021  =  .4.  7.   antilog  2.5221  =  .03328. 
Find  the  antilogarithm  of : 

8.  2.9605.  10.   1.8451.  12.   6.4983. 

9.  0.5963.  11.   1.8401.  13.   8.0755-10, 


338  HIGH  SCHOOL  ALGEBRA  [Ch.  XXII 

14.  3.3997.  18.  3.7361.  22.  1.3019. 

15.  4.2226.  19.  0.9002.  23.  5.9754-10. 

16.  2.6512.  20.  2.9068.  24.  9.5327-10. 

17.  1.8846.  21.  5.8049.  25.  4.6831  -  10. 

218.   Computation  by  means  of  logarithms. 
Ex.  1.   Find  p,  if  j>  =  47.45  x  3.514  x  .0064. 

SOLUTION 

log  p  =  log  47.45  +  log  3.514  +  log  .0064 ;    [§  212,  Prin.  2 
but  log  47.45=    1.6763,  [§216 

log  3.514=    0.5458, 
and  log  .0064  =    7.8062  -  10,^      [§  216,  Note 

therefore  log  p  =  10.0283  -  10 

=    0.0283; 
and  therefore  p  =    1.067.  [§  217 

This  product  found  in  the  ordinary  way  is  .10671+. 

Ex.  2.   Find  3.041^ 

Solution,     log  (3.041^)  =  4  x  log  3.041  [§  212,  Prin.  4 

=  4  X  0.4830  =  1.9320 ;  [§  216 

therefore  3.041^  =  antilog  1.9320  =  85.5.  [§  217 

Obtained  by  ordinary  multiplication  3.041*  =  85.5196+. 
Ex.  3.   Find  ^:0572. 


Solution,     log  V.0572  =  ^  x  log  .0572  [§  212,  Prin.  5 


=  I  X  2.7574 

=  ix  (1.7574 -3) t 

=  0.5858-1  =  1.5858; 


therefore  V.0572  =  antilog  1.5858  =  .3853. 

Obtained  by  the  method  of  §  120,  -^;0572  =  .38529+. 

*  The  form  7.8602  -  10  (instead  of  3.8062)  is  used  for  log  .0064  because, 
in  computation,  negative  characteristics  increase  the  danger  of  errors. 

t  In  order  to  divide  2.7574  by  3  without  mixing  positive  and  negative 
numbers  it  is  well  first  to  write  2.7574  in  one  of  the  following  forms: 
1.7574  -  3,  4.7574  -  G,  7.7574  -  9,  etc.,  i.e.,  to  add  (and  then  subtract)  some 
multiple  of  3  which  will  make  the  characteristic  positive. 


217-218] 


LOGARITHMS 


339 


„     ,    J.        37.22  X  (-19.86)    n    , 
'^•*-  ^'^= (12.33y        -^"^"' 

Solution.  In  such  examples  we  first  find  the  numerical  value 
of  the  result  by  regarding  all  the  factors  as  positive,  and  then 
prefix  the  proper  sign  as  determined  by  §§  18  and  19.  Thus, 
ignoring  the  minus  sign,  we  have 

log  X  =  log  137.22  +  log  9.86  -  2  x  log  12.33   [§  212,  Prin.  2  and  3 
=  1.5707  +  1.2980  -  2  x  1.0910 
=  0.6847  ; 

therefore  x=-  antilog  0.6847  =  - 4.84. 

Ex.  5.    Given  47.5^  =  293.64  ;  find  x. 

Solution.  On  taking  the  logarithm  of  each  member  of  this 
equation  we  obtain 

X .  log  47.5  =  log  293.64 
^  log  293.64  . 
log  47.5    ' 
2.4678 


whence 


I.e.. 


x  = 


1.6767 


1.472. 


Note.  Equations  in  which  the  unknown  number  appears  as  an  exponent 
are  called  exponential  equations.  Such  equations  cannot  be  solved  by  the 
methods  given  in  the  preceding  pages,  but  are  easily  solved  by  the  method 
illustrated  in  the  above  solution  of  Ex.  5. 


EXERCISE  CXLVIII 
By  logarithms  find  the  value  of  : 

6.  376x58.  12.  380.7 -^  9.8. 

7.  2.29x8.7.  13.  10 -^  3.141. 

8.  69.5  x. 00543.  14.  3 -r- 5.963. 

9.  -42.37  X. 236.  15.  30.07 -?- .002121. 

10.  .2912x3.141.  16.  .005918 -f- .0009293. 

11.  .0695  x  .002682.  17.  13  x  753  ^  .06238. 


340  HIGH  SCHOOL  ALGEBRA  [Ch.  XXII 

By  logarithms  simplify : 

18.  23\  23.    (2)8.  28.  V675. 

19.  .08^1  24.   (If)^.  29.  ^:0500l. 

20.  .395^-1^  25.   (62)i  30.  ^(.3192)«. 


21.   (-3.813)'.  26.   (991.7)^.  3^^   -;/:i277^l7. 

32.  '^18^V2574. 


22.   (1.228)10.  27.   (.1183)? 


33. 
34. 


19x(-700)  4635^«  X  200.4* 


970  X  1.4  X  .0616 


36. 


10123 


3-1^1  X  .0711  ^^    13^n^2^5 

.8331x51  •      57o^7)j2l 

33^       1.78  X. 0052x16.                         2^x(#X^f 
^     .339x4.315  38.    ^2J^ ^. 

39.  If  a,  b,  and  c  are  the  sides  of  a  triangle,  and  s  is  one  half 
their  sum,  the  area  of  the  triangle  is  -\/s(s —  a)(s  —  b)(s  —  c). 
Find,  by  logarithms,  the  area  of  the  triangle  whose  sides  are 
13.6  ft.,  15.1  ft.,  and  20.1  ft. ;  also  the  area  of  the  triangle  whose 
sides  are  260  ft.,  319  ft.,  and  464  ft. 

Solve  for  x  (cf.  Ex.  5) : 

40.  16-"=  354.  43.   6"^  =  5'=+\ 

41.  7^  =  9.59.  44.  2'^  =  113-+!. 

42.  28.8^  =  12750.  45.   152»<^«^  =  3275. 

46.  From  (1),  §  201,  show  that  in  a  G.  P.  log  r  =  ]2K1zl}^.^ 

n  —  1 
also  find  r  when  a  =  10,  w  =  10,  and  I  =  196830. 

47.  If  A  is  the  amount  of  P  dollars  at  r  %  compound  interest 
for  n  years,  show  that  A  =  P(l  +  r)" ;  also  solve  this  equation  for 
each  letter  it  contains.     (Cf.  Ex.  25,  p.  316,  also  Ex.  46  above.) 

48.  Find  the  amount  of  $700  for  5  years  at  4%  compound 
interest;  also  the  amount  of  $450  for  10  years  at  3%  compound 
interest. 

49.  In  what  time  will  $  800  amount  to  $  1834.50  if  put  at  com- 
pound interest  at  5  %  ? 


218] 


LOGARITHMS 


341 


Table  of  Common  Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

CXXX) 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1 106 

13 

1 139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

I46I 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

I76I 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

i6 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3(>3^ 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43H 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

477^ 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5159 
5289 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5|3? 

555' 

36 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

^Hl 

5888 

5899 

39 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

^?'^ 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

?Zt4 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691 1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7H3 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

73Sf 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

342 


HIGH    SCHOOL    ALGEBRA 


Table  of  Common  Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

826i 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175' 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

98CX) 

9805 

9809 

9814 

9818 

96 

9827 

9832 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

INDEX 


[Numbers  refer  to  pages.] 


Absolute,  term,  190. 

value,  19. 
Addition,  8,  21,  29,  30,  115,  243. 
Algebraic,  expressions,  27. 

fraction,  109. 

numbers,  19. 

sentence,  60. 

sum,  21. 
Alternation,  298. 
Antecedent,  293. 
Antilogarithm,  ,336, 
Approximate  root,  181. 
Arithmetical,  mean,  311. 

numbers,  19. 

progression,  307. 

series,  307. 
Arranged  polynomial,  43. 
Associative  law,  54. 
Axioms,  56. 
Axis,  of  coordinates,  219. 

of  imaginaries,  259. 

of  real  numbers,  259. 

Base,  of  logarithms,  329. 

of  power,  12. 
Binomial,  27,  236. 
,     cube  of,  76. 

square  of,  71. 

theorem,  174,  323. 
Brace,  bracket,  etc.,  15. 
Briggs  logarithms,  332, 

Character  of  roots,  278. 
Characteristic  of  logarithm,  333. 
Checking  results,  31,  56. 
Clearing  of  fractions,  126. 
Coefficients,  27,  190,  236. 


Commensurable  numbers,  293. 
Common,  difference,  307. 

factor,  98, 

logarithms,  332, 

multiple,  105. 

ratio,  313. 
Commutative  law,  54. 
Completing  the  square,  194. 
Complex,  fractions,  123. 

numbers,  257. 
Composite  numbers,  77. 
Conditional,  equation,  55. 

inequality,  289. 
Conjugate,  complex  numbers,  257. 

surds,  249. 
Consequent,  293. 
Consistent  equations,  146. 
Constant  term,  190. 
Constants,  301. 
Continued,  product,  25. 

proportion,  296. 
Coordinate  axes,  219. 
Cube,  13. 

of  a  binomial,  76. 

root,  176,  185,  187. 
Cubic  equation,  126. 

Decreasing  series,  316. 
Degree,  of  term,  43. 

of  equation,  125. 
Denominator,  109. 
Determinate  system,  165. 
Difference,  8. 
Discriminant,  278. 
Dissimilar,  radicals,  236. 

terms,  28. 
Distributive  law,  54. 
343 


U4: 


INDEX 


Division,  11,  26,  45,  121,  250,  298. 
Divisor,  dividend,  12. 

Elements   of  A.  P.    and   G.  P.,  307 

313. 
Elimination,  147,  149. 
Equations,  conditional,  55. 

consistent,  146. 

determinate,  165. 

equivalent,  127. 

exponential,  339. 

fractional,  130,  151,  198. 

graph  of,  221. 

homogeneous,  208. 

identical,  55. 

inconsistent,  146. 

independent,  146. 

indeterminate,  144,  165. 

in  quadratic  form,  205. 

integral,  125. 

irrational,  255. 

linear,  125. 

literal,  125,  128,  154. 

locus  of,  221. 

numerical,  125. 

of  tlie  problem,  62. 

quadratic,  126,  190,  277. 

radical,  255. 

simple,  125. 

simultaneous,  146,  206. 

solution  of,  57. 

symmetric,  212. 
Equivalent  equations,  127. 
Even,  power,  170. 

root,  176. 
Evolution,  176. 
Exponent,  12. 

fractional,  negative,  zero,  265 
laws,  38,  45,  171,  265. 
Exponential  equation,  339. 
Extraneous  roots,  131. 
Extremes,  296,  311,  318. 


Factoring,  78,  90,  282. 

solving  equations  by,  94. 
Factors,  10,  77,  78. 

of  quadratic  expressions,  282. 
Factor  theorem,  92. 


Finite,  numbers,  52. 

series,  307. 
Formulas,  for  A.  P.,  G.  P.,  308,  314. 

for  solving  equations,  140,  277. 
Fourth  proportional,  296. 
Fractional,  equations,  130,  151. 

exponent,  265. 
Fractions,  12,  109. 

clearing  of,  126. 

lowest  terms  of.  111. 

General  problem,  139. 
Geometric,  infinite  G.  P.,  316. 

means,  318. 

series,  313. 
Graph  of  an  equation,  221. 
Graphic  solutions,  227. 
Graphical  representation  of  complex 

numbers,  259. 
Greater  than,  287. 

Harmonic  series,  319. 
Highest  common  factor,  98. 
Homogeneous  equations,  208. 


Identical  equations,  55. 
Imaginary,  numbers,  177,  235,  257. 

unit,  257. 
Improper  fraction,  109. 
Incommensurable  numbers,  293. 
Inconsistent  equations,  146. 
Independent  equations,  146. 
Indeterminate,  equations,  144,  165. 

systems,  165. 

Index  of  a  root,  176. 

Induction,  mathematical,  322. 

^  Jnequalities,  287. 

-^Infinite  series,  307,  316. 

Infinitely,  large,  52. 

small,  52. 
Insertion  of  parentheses,  35. 
Integral,  equation,  125. 

expressions,  42. 
Interpretation,  of  results,  189. 


of  the  forms 


«-,  .5,  301, 


0     00     0 

Inverse,  operations,  9. 
ratio,  293. 


INDEX 


345 


Inversion  of  proportion,  297. 
Involution,  170. 
Irrational,  equation,  255. 
numbers,  235. 

Known  and  unknown  numbers,  125. 

Laws,  of  exponents,  38,  45,  171,  265. 

of  operations,  54. 

of  signs,  25,  26,  177. 
Less  than,  287. 
Letter  of  arrangement,  43. 
Like,  and  unlike,  radicals,  236. 

terms,  28. 
Limit,  301. 

Linear  equations,  125. 
Literal,  coefficients,  27. 

equations,  125,  128,  154. 

numbers,  1,  3, 
Locus  of  an  equation,  221. 
Logarithms,  329. 

table  of,  341. 
Lowest  common  multiple,  105. 

Mantissa  of  logarithm,  333. 
Mathematical  induction,  322. 
Mean  proportional,  296. 
Members  of  an  equation,  55. 
Minuend,  9. 
Mixed  expression,  109. 
Monomials,  27. 
Multiples,  105. 
Multiplicand,  multiplier,  10. 
Multiplication,  10,  38. 

Negative,  exponent,  266. 

numbers,  18. 

term,  28. 
Numbers,  absolute  value  of,  19. 

commensurable,  etc.,  293. 

complex,  257. 

constants  and  variables,  301. 

finite  and  infinite,  52. 

imaginary,  177,  235,  257. 

known  and  unknown,  125. 

literal,  1,  3. 

negative  and  positive,  18. 

opposite,  19. 


Numbers,  prime  and  composite,  77. 

rational  and  irrational,  234,  235. 

real,  177,  235. 
Numerical,  coefficient,  27. 

equation,  125. 

Odd,  power,  170. 

root,  176. 
Operations,  with  literal  numbers,  1. 

with  imaginary  and  complex  num- 
bers, 261. 
Opposite,  numbers,  19. 

species,  287. 
Order,  of  operations,  14. 

of  radicals,  236. 

Parentheses,  15,  35. 
Polynomials,  27. 

square  of,  75,  328. 
Positive  numbers,  18. 

terras,  28. 
Power,  12. 

Powers  of  imaginary  unit,  258. 
Prime,  numbers,  77. 

to  each  other,  98. 
Principal  roots,  236. 
Principles,  of  clearing  of  fractions, 
126. 

of  elimination,  147,  149. 

of  inequalities,  287. 

of  logarithms,  303. 

of  proportion,  296. 
Problems,  62. 

directions  for  solving,  62. 

general,  139. 
Products,  10,  24,  38,  40,  etc. 

of  fractions,  119. 

of  sum  and  difference,  72. 
Progression,  arithmetical,  307. 

geometric,  313. 

harmonic,  319. 
Proof  by  induction,  321. 
Proper  fraction,  109. 
Property,  of  complex  numbers,  262. 

of  quadratic  surds,  253. 
Proportion,  295. 
Pure,  quadratic,  190. 

imaginary  numbers,  257. 


346 


INDEX 


Quadratic  equation  has  two  roots,  and 

only  two,  283. 
Quadratic  equations,  126,  190,  277. 

form  of,  205. 

graphs  of,  229. 

roots  of,  278,  279, 

simultaneous,  206. 

solution  by  formula,  277. 

special  devices  for,  211. 

surds,  253. 
Quotient,  12. 

Radicals,  radical  equations,  235,  236, 

241,  255. 
Radicand,  235. 
Ratio,  293,  294,  313. 
Rational  numbers,  234. 
Rationalizing  factor,  249,  276. 
Real  numbers,  177,  235. 
Reciprocal  of  a  number,  110. 
Relation    between   roots   and  coeffi- 
cients, 279. 
Remainder,  8. 

theorem,  92. 
Removal  of  parentheses,  35. 
Review  exercises,  67,  166,  284. 
Root,  principal,  236. 

to  n  terms,  181. 
Roots,  of  an  equation,  56. 

character  of,  278. 

extraneous,  131. 

relation  between  coefficients  and, 
279. 
Rule  of  signs,  25,  26. 

Series,  307. 

Signs,  of  aggregation,  14. 

of  deduction,  3. 

of  inequality,  287. 

of  operation,  1 ,  10,  19. 

of  quality,  19. 
Similar,  radicals,  236. 

terms,  28. 
Simple  equations,  125. 
Simultaneous  equations,  146,  206. 
Solution  of  equations,  57,  146,  206,  etc. 

by  factoring,  94. 


Species  of  inequalities,  287. 
Specitic  gravity,  139. 
Square,  13. 

of  a  binomial,  71. 

of  a  polynomial,  75,  328. 
Square  root,  176,  180,  188. 

of  binomial  surd,  253. 

of  complex  number,  262. 
Standard  form,  of  complex  number, 
257. 

of  quadratic,  190. 
Subtraction,  8,  22,  32. 
Subtrahend,  8. 
Sum,  8,  21. 
Summands,  8. 
Surds,  235. 

conjugate,  249. 
Symbols,  •.•  and  .-.,3, 

>  and  <,  287. 
Symmetric  equations,  212. 
System  of  equations,  146. 

indeterminate,  165. 

Table  of  logarithms,  341. 

Terms,  27, 109,  203,  307. 

Theorem,  binomial,  174,  323. 

Thermograph,  232. 

Third  proportional,  296. 

Translation  of  common  language  into 

algebraic  language,    and    vice 

versa  ^  60. 
Transposing,  58. 
Trinomial,  27. 
Type  forms,  71. 

Unconditional  inequality,  289. 
Unknown  numbers,  125. 
Unlike  terms,  28. 

Variable,  variation,  301. 
Vary,  304. 

directly,  304. 

inversely,  304. 

jointly,  304. 
Vinculum,  15. 

Zero,  52. 

exponents,  266. 


^N  INITIAL  FINE  OF  25  CEKTS 

OVERDUE.  ■= 


I,D2l-100rrv-7,'40  (6936s) 


T3^ 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


